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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\mathcal O\) be the local ring at a singularity of a geometrically integral algebraic curve defined over a finite field \(\mathbb F_q\), and let \(m\) be the number of branches centered at the singularity. In a previous paper the second author extended the notion of partial local zeta-functions, by considering for each pair of \(\mathcal O\)-ideals \(\mathfrak a\) and \(\mathfrak b\) a Poincaré series \(P(\mathfrak {a,b},t_1,\dots,t_m)\) in \(m\) variables, which encodes cardinalities of certain finite sets of ideals. To study the behavior of these power series under blow-ups, we generalize the theory by allowing that \(\mathcal O\) is a semilocal ring of the curve. In this context we establish an Euler product identity, which provides the connection between the local and semilocal theory. We further present a procedure to compute the Poincaré series, and illustrate the method by some examples of local rings. Another purpose of this paper is to study the reduction \(\mod q-1\) of \(P(\mathfrak{a,b},t_1,\dots,t_m),\) which becomes a polynomial if \(m>1\). Poincaré series; local rings; finite fields Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Singularities of curves, local rings, Computational aspects of algebraic curves On Poincaré series of singularities of algebraic curves over finite fields
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The present paper deals with the arithmetic Bloch-Ogus complex for 2-dimensional complete normal local rings of positive characteristic whose residue field is a local field. More precisely, let \(A\) be such a ring, \(K =\) Frac \((A),\) \(k =\) the residue field, \(X=\text{ Spec}(A)\setminus \{\)maximal ideal of \(A \}.\) The author shows a ``duality theorem'': For any prime number \(\ell\) different from Char \(A,\) one has an isomorphism \(H^5(X, {\mathbb Z}/\ell (3)) \simeq {\mathbb Z}/\ell\) and a perfect duality
\[
H^1(X, {\mathbb Z}/\ell) \times H^4(X, {\mathbb Z}/\ell (3)) \to H^5(X, {\mathbb Z}/\ell (3)) \simeq {\mathbb Z}/\ell
\]
One first consequence, by a ``purity'' result of Fujiwara-Gabber, is an exact sequence:
\[
0 \to \pi_1^{c,s}(X)/\ell \to H^4 (K, {\mathbb Z}/\ell(3)) \to {\mathop\bigoplus_{v \in P}} H^3(k(v), {\mathbb Z}/\ell(2)) \to {\mathbb Z}/\ell \to 0,
\]
where \(\pi_1^{c,s}(X)\) denotes the quotient group of \(\pi_1^{ab}(X)\) which classifies abelian ``completely split'' coverings of \(X,\) \(P\) is the set of height one prime ideals of \(A,\) \(k(v)\) the residue field at \(v\) of the henselization of \(A.\) For \(A = {\mathbb F}_p ((t)) [[X, Y]]\) (a regular ring), using a recent result of \textit{I.A. Panin} [Proc. Steklov Inst. Math. 241, 154--163 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 169--178 (2003; Zbl 1115.19300 )] on the homology of the Bloch-Ogus complex, one can then show the vanishing of \(\pi_1^{c,s}(X)/\ell,\) hence a cohomological Hasse principle for \({\mathbb F}_p((t)) [[X, Y]].\) (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Geometric class field theory, Curves over finite and local fields, Global ground fields in algebraic geometry, Generalized class field theory (\(K\)-theoretic aspects), Étale and other Grothendieck topologies and (co)homologies Cohomological Hasse principle for the ring \(\mathbb F_p((t))[[X, Y]]\)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this work, authors provide an algebro-geometric approach to Vakil-Zinger's desingularization of the main compartment of moduli space of genus one stable maps to \(\mathbb{P}^{n}\).
Vakil-Zinger found a canonical desingularization of \(\overline{\mathcal{M}}_{1}(\mathbb{P}^{n},d)_{0}\) via virtual blowing-ups. Although these results are algebro-geometric, the proof is analytic. The authors provide an algebro-geometric approach to these desingularization results. This will also serve as their first step to generalize results on moduli spaces of genus one stable maps to higher genera. stable maps; local equations; Vakil-Zinger's desingularization; Gromov-Witten invariants; virtual blowing-up Hu, Y; Li, J, Genus-one stable maps, local equations, and vakil-zinger's desingularization, Math. Ann., 348, 929-963, (2010) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Global theory and resolution of singularities (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Genus-one stable maps, local equations, and Vakil-Zinger's desingularization
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For the entire collection see Zbl 0742.00073.]
The aim of this paper is to extend to certain singular varieties the results of \textit{S. Bloch} and \textit{A. Ogus} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7(1974), 181-201 (1975; Zbl 0307.14008)] for the Zariski cohomology groups \(H^ p(X,{\mathcal H}^ q)\), \({\mathcal H}^ q\) being a suitable cohomology theory sheaf on the variety \(X\) (e.g. étale cohomology). We prove the existence of a resolution for the sheaf \({\mathcal H}^ q\) on \(X\), analogous to that given in the paper cited above for nonsingular varieties. When \(X\) is singular this resolution is, in general, not acyclic: the obstruction is computed by cohomology groups \(H^*(X_ Y,{\mathcal H}^ q)\) where \(X_ Y\) is the localized scheme of \(X\) at the singular locus \(Y\) (we assume \(Y\) is contained in an affine open subset of \(X)\). Then (\S3) we apply these results to étale cohomology and relate the groups \(H^ n(X,{\mathcal H}^ n_{\text{ét}})\), where \(n=\dim X\), with the relative Chow group of zero cycles on \(X\). resolution for cohomology sheaf; singular varieties; Zariski cohomology groups; étale cohomology; Chow group; zero cycles Applications of methods of algebraic \(K\)-theory in algebraic geometry, Singularities in algebraic geometry, Algebraic cycles, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Cohomology theories and algebraic cycles on singular varieties
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Given distinct points \(p_1,\dots, p_r\) of the projective plane \(\mathbb{P}^2\) over an algebraically closed field, a fat point subscheme \(Z= m_1p_1 +\cdots+ m_rp_r\) of \(\mathbb{P}^2\) is the subscheme defined by an ideal of the form \(I_Z= {\mathfrak p}_1^{m_1} \cap\cdots\cap {\mathfrak p}_r^{m_r}\), where \({\mathfrak p}_i\) is the prime ideal corresponding to \(p_i\). By allowing infinitely near points, the author extends this notion, and then tries to compute the graded free resolution of \(I_Z\) in case \(p_1,\dots, p_r\) are contained in a plane curve of degree \(\leq 3\). Using the blowing up \(X\) of \(\mathbb{P}^2\) at \(p_1,\dots, p_r\) and the sheaf \({\mathcal F}_d\) corresponding to the divisor \(de_0- m_1e_1-\cdots- m_re_r\) for \(d\in\mathbb{Z}\), the problem amounts to computing \(h^0(X,{\mathcal F}_d)\) and the dimension \(s({\mathcal F}_d, e_0)\) of the cokernel of \(H^0(X, {\mathcal F}_d)\otimes H^0(X, e_0)\to H^0(X,{\mathcal F}_{d+1})\). Thus the problem splits into finding the monoid of effective divisor classes, their Zariski decompositions, and \(h^0(X,{\mathcal F})\) as well as \(s({\mathcal F}, e_0)\) for all \({\mathcal F}\) in the cone of numerically effective classes.
In the case of points on a line or conic, the author shows \(s({\mathcal F}, e_0)= 0\) for all numerically effective classes, whereas in the case of points on a cubic his results are less comprehensive and rely on certain restrictions on the classes \({\mathcal F}\) or the surfaces \(X\) under consideration. fat point subscheme; infinitely near points; effective divisor classes Harbourne, B.: Free resolutions of fat point ideals on P\^{}\{2\}. J. Pure Appl. Algebra. 125(1-3), 213-234 (1998) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Computational aspects and applications of commutative rings Free resolutions of fat point ideals on \(\mathbb{P}^2\)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be an algebraically closed field of characteristic zero, \(R= K[[x_1, \dots x_n]]\) be the formal power series ring and \(A = R/I\) a local ring with maximal ideal \(m = (x_1, \dots, x_n)/I\). The algebra \(A\) is said to be canonically graded if there exists a \(K\)-algebra isomorphism between \(A\) and its associated graded ring \(\mathrm{gr}_m(A) = \bigoplus_{i \geq 0} m^i/m^{i+1}\). Throughout the paper, \(A=R/I\) is an Artin local ring. The socle of \(A\) is the ideal defined by \(\mathrm{Soc}(A):= (0:m)\), and the socle degree of \(A\) is the maximum integer \(j\) such that \(m^j \neq 0\). Let \(t:=\dim_K \mathrm{Soc}(A)\) be the type of \(A\). If \(A\) is of initial degree \(v\) i.e \(I \in (x_1,\dots, x_n)^v \backslash (x_1, \dots, x_n)^{v+1}\) and of socle degree \(s\), then the socle type \(E\) of \(A\) is defined to be \(E= (0, \dots, e_{v-1}, e_v, \dots, e_s, 0, \dots, 0)\) where
\[
e_i = \dim_K(\mathrm{Soc}(A) \cap m^i/\mathrm{Soc}(A) \cap m^{i+1})
\]
A level algebra \(A\) of socle degree \(s\), type \(t\) and embedding dimension \(n\) is an Artin quotient of \(R\) whose socle type is of the following form \((0, \dots, 0, e_s,0, \dots 0)\) with \(e_s =t\). The Artin algebra is Gorenstein if \(t=1\).
An Artin algebra \(A = R/I\) of socle type \(E\) is said to be compressed if it has maximal length \(e(A) = \dim_K A\) among all the Artin quotients of \(R\) having socle type \(E\) and embedding dimension \(n\).
In the paper under review, the authors prove in theorem 3.1 that any Artin compressed Gorenstein local \(K\)-algebra of socle degree \(s \leq 4\), is canonically graded. They also show in theorem 3.2 that an Artin compressed \(K\)-algebra \(A\) of embedding dimension \(n\), socle degree \(s\) and socle type \(E\) is canoncally graded in any of the following cases: {\parindent=6mm \begin{itemize}\item[1)] \(s \leq 3,\) \item[2)] \(s=4\) and \(e_4 =1,\) \item[3)] \(s=4\) and \(n=2.\)
\end{itemize}} Further, they give explicit examples to show that these results do not work in general. Examples 3.3 and 3.4 show that theorem 3.1 fails when \(A\) is Gorenstein of socle degree \(4\) but not compressed or when \(A\) is compressed Gorenstein of socle degree \(5\). As well, example 3.5 shows that theorem 3.2 cannot be extended to socle degree \(4\) and type \(>1\). local ring; canonically graded; associated graded ring; socle; socle degree; socle type; type; level algebra; compressed algebra; Gorenstein algebra Elias, J; Rossi, ME, Analytic isomorphisms of compressed local algebras, Proc. Am. Math. Soc., 143, 973-987, (2015) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Multiplicity theory and related topics, Parametrization (Chow and Hilbert schemes) Analytic isomorphisms of compressed local algebras
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\) be the coordinate ring of a finite union of subspaces of an affine space and \(S\) be its seminormalization. If \(X\) is the corresponding intersection poset and \(\mathcal A\) the sheaf of coordinate rings on \(X\), then the author shows that \(S\) is Cohen-Macaulay if and only if \(\mathcal A\) is ``gradual'' and for every \(x\in X\), \(\tilde H^i(X^x,\mathcal A)=0\) for \(i< \mathrm{rk}(x)-2\), where \(X^x:=\{y\in X\mid y<x\}\). The main idea is to use a result of Orecchia that expresses \(S\) as \(\Gamma(\mathcal A)\), then to estimate the depth of \(S\) by constructing a resolution by modules of known depth. The result generalizes an equivalent version of Reisner's theorem on the Cohen-Macaulay property of face rings of simplicial posets. unions of linear subspaces; coordinate ring; seminormalization; intersection poset; Cohen-Macaulay; face rings [Y3] Yuzvinsky, S.: Cohen-Macaulay seminormalizations of unions of linear subspaces. J. Algebra132, 431--445 (1990) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Seminormal rings, Relevant commutative algebra Cohen-Macaulay seminormalizations of unions of linear subspaces
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Maps between \(k\)-algebras \(f:B\rightarrow C\) are considered to be [\textit{first-order infinitesimal}] \textit{neighbours} if \((f(a)-g(a))(f(b)-g(g))=0\) for all \(a,b\). If \(2\in k\) is invertible, this is equivalent to \((f(a)-g(a))^2\) being zero. This paper gives an equivalent formulation of this property in terms of ideals, and uses this to study the neighbour property where \(B\) is a polynomial algebra.
A \((p+1)\)-tuple of mutually neighbouring algebra maps is defined to be an \textit{infinitesimal \(p\)-simplex}. It is shown that affine combinations of such algebra maps are themselves algebra maps (which is not true in the general case), and that two such affine combinations are themselves neighbours. Results are extended to vectors with entries from \(C\), and to vector spaces over a ring of scalars. first neighbourhood of the diagonal; neighbour points; affine schemes; affine combinations Infinitesimal methods in algebraic geometry, Formal neighborhoods in algebraic geometry, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Synthetic differential geometry Affine combinations in affine schemes
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(k\) be a field and \(A^n_k\) be the \(n\)-dimensional affine space over \(k\). The following two results are known to be fundamental in the study of torsors over \(k\):
(1) \(H^1_{\mathrm{Zar}}(A^n_k,\mathrm{GL}_n)=H^1_{\text{ét}}(A^n_k,\mathrm{GL}_n)=H^1_{\mathrm{fppf}}(A^n, \mathrm{GL}_n)\) and
(2) \(H^1_{\mathrm{Zar}}(A^n_k,\mathrm{GL}_n)=1\).
The first of these two results, a version of Hilbert 90, asserts that the Zariski topology - which is in general too coarse to deal with principal bundles - is fine enough to measure algebraic vector bundles. The second is equivalent to the theorem of Quillen and Suslin: All finitely generated projective modules over the polynomial ring \(k[t_1,t_2,\ldots,t_n]\) are free.
The main objective of the paper under review is to look at the above two questions in the case when \(k[t_1,t_2,\ldots,t_n]\) is replaced by the ring \(R_n=k[t_1^{\pm}{}^,\ldots,t_n{}^{\pm 1}]\) of Laurent polynomials in \(n\)-variables and \(\mathrm{GL}_n\) is replaced by some other affine smooth group scheme over \(R_n\). Assume \(k\) is algebraically closed of characteristic 0. Consider the class of Lie algebras \(L\) over \(R_n\) with the property that \(L\otimes_{R_n}S\simeq g\otimes_k S\) for some finite dimensional simple Lie algebra \(g\), and some cover \(S\) of \(R\) on the étale topology. Observe that in this way (for \(n=1\)) one obtains the affine Kac-Moody Lie algebras. All these algebras are parametrised by \(H^1_{\text{ét}}(R,\Aut(g))\), namely by torsors over \(R\) under \(\Aut(g)\). The main result of the paper is that any such torsor is isotrivial, i.e., trivialized by a finite étale extension of \(R_n\). Gille, Philippe and Pianzola, Arturo Isotriviality and étale cohomology of Laurent polynomial rings \textit{J.~Pure Appl. Algebra}212 (2008) 780--800 Math Reviews MR2363492 (2009c:14089) Group schemes, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Linear algebraic groups over arbitrary fields Isotriviality and étale cohomology of Laurent polynomial rings
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system If \(G\) is a reductive group acting on a linearized smooth scheme \(X\) then we show that under suitable standard conditions the derived category \(\mathcal{D}(X^{ss}/G)\) of the corresponding GIT quotient stack \(X^{ss}/G\) has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on \(X^{ss}\sslash G\) which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of \(X^{ss}\sslash G\) constructed earlier by the authors. As a concrete example we obtain in the case of odd Pfaffians a semi-orthogonal decomposition of the corresponding quotient stack in which all the parts are certain specific non-commutative \textit{crepant} resolutions of Pfaffians of lower or equal rank which had also been constructed earlier by the authors. In particular this semi-orthogonal decomposition cannot be refined further since its parts are Calabi-Yau. The results in this paper complement results by Halpern-Leistner, Ballard-Favero-Katzarkov and Donovan-Segal that assert the existence of a semi-orthogonal decomposition of \(\mathcal{D}(X/G)\) in which one of the parts is \(\mathcal{D}(X^{ss}/G)\). non-commutative resolutions; geometric invariant theory; semi-orthogonal decomposition Geometric invariant theory, Noncommutative algebraic geometry, Derived categories and associative algebras Semi-orthogonal decompositions of GIT quotient stacks
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be an algebraically closed field. Let \(P\subseteq K[x_0, \dots, x_n]\) be a homogeneous maximal ideal and \(S\) the homogeneous subalgebra of \(K[x_0, \dots, x_n]\) generated by the linear forms of \(P\). Let \(I\subset K[x_0, \dots, x_n]\) be a homogeneous ideal and \(y\) a linear form not in \(P\). The \(k\)-th partial elimination ideal of \(I\) with respect to \(P\) is given by
\[
K^P_k(I)={\bigoplus}_{d\in\mathbb{Z}}\{f\in S_d\mid\exists g\in (P^{d+1})_{d+k}:y\;^k\!f+g\in I_{d+k}\}.
\]
It is proved that the cone of \((k+1)\)-secant lines of a closed subscheme \(Z\subseteq\mathbb{P}^n_k\) through a closed point \(p\in\mathbb{P}^n_k\) is defined by the \(k\)-th partial elimination ideal of \(Z\) with respect to \(p\). This is the bases for an algorithm to compute secant cones. Examples are studied using the computer algebra system \textsc{Singular}. partial elimination ideal; secant cone; secant locus; linear projection Kurmann, S, Partial elimination ideals and secant cones, J. Algebra, 327, 489-505, (2011) Computational aspects of higher-dimensional varieties, Computational aspects and applications of commutative rings, Software, source code, etc. for problems pertaining to algebraic geometry Partial elimination ideals and secant cones
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system For a linear system \(L\) on a projective variety \(X\) and \(r\) general points with multiplicities it is a classical question whether the latter impose independent conditions on \(L\). The paper generalizes this question in two directions: first the author considers the situation where the system of effective divisors in \(L\) passing through the points with given multiplicities fails to have the expected dimension by a fixed positive value; then he replaces the \(r\) general points by \(r\) general lines. Projective techniques in algebraic geometry Families of fat points.
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(k\) be an algebraically closed field of characteristic zero. The authors generalize the Cayley \(\Omega\)-processes, constructing a (generalized) \(\Omega\)-process on any linear algebraic monoid \(M\) over \(k\) with zero, a dense unit group \(G\) and a nontrivial character \(\lambda\), as a nonzero linear operator \(\Omega\colon k[M]\to k[M]\) such that for all \(f\in k[M]\) and \(m\in M\), \(\Omega(f\cdot m)=\lambda(m)\Omega(f)\cdot m\) and \(\Omega(m\cdot f)=\lambda(m)m\cdot\Omega(f)\). The authors also show how to produce a number of elements of the ring of \(G\)-invariants \(S(V)^G\) that is large enough to guarantee its finite generation from the process and for a rational representation of \(G\). Moreover, using complete reducibility, they give an explicit construction of all \(\Omega\)-processes for reductive monoids. Cayley \(\Omega\)-process; characters; linear algebraic monoids; polynomial representations; rational representations; reductive monoids; rings of invariants Santos, W. Ferrer; Rittatore, A.: Generalizations of Cayley's \({\Omega}\)-process, Proc. amer. Math. soc. 135, No. 4, 961-968 (2007) Semigroups of transformations, relations, partitions, etc., Linear algebraic groups over arbitrary fields, Geometric invariant theory, Actions of groups on commutative rings; invariant theory Generalizations of Cayley's \(\Omega\)-process.
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\) be a variety embedded in the projective space \(\mathbb{P}^N\) over an algebraically closed field \(k\). We denote by \(\pi_z: \mathbb{P}^N\setminus\{z\}\to \mathbb{P}^{N-1}\) the projection from a point \(z\in\mathbb{P}^N\) and define
\[
\begin{aligned} {\mathfrak S}^{\text{out}}(X) &:= \overline{\{z\in \mathbb{P}^N\setminus X\mid\pi_{z|X}: X\to\pi_z(X)\text{ is not birational}\}},\\ {\mathfrak S}^{\text{inn}}(X) &:= \overline{\{z\in X\mid \pi_{z|X}: X\setminus\{z\}\to \pi_z(X\setminus\{z\})\text{ is not birational}\}}.\end{aligned}
\]
The first set is called the Segre locus of \(X\) in regard to an old theorem due to B. Segre who proved that \({\mathfrak S}^{\text{out}}(X)\) is a finite union of linear subspaces of \(\mathbb{P}^N\), provided \(k\) has characteristic zero. The set \({\mathfrak S}^{\text{tot}}(X):={\mathfrak S}^{{out}}(X)\cup{\mathfrak S}^{\text{inn}}(X)\) is called the total Segre locus of \(X\). We quote the author's main result:
Theorem. Assume that \(X\subset\mathbb{P}^N\) is a nondegenerate projective (reduced and irreducible) variety over an algebraically closed field of characteristic \(p\). If either \(p\geq\deg(X)\) or \(p= 0\), then the total Segre locus \(S^{\text{tot}}(X)\) of \(X\) is a finite union of linear subspaces of \(\mathbb{P}^N\).
The theorem implies Segre's result. Furthermore, the authors give an example which shows that the assertion of the theorem does not hold in general, if \(p< \deg(X)\). Segre locus Furukawa, Katsuhisa, Defining ideal of the Segre locus in arbitrary characteristic, J. Algebra, 0021-8693, 336, 84-98, (2011) Projective techniques in algebraic geometry, Determinantal varieties Defining ideal of the Segre locus in arbitrary characteristic
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R, \mathfrak{m}, k)\) be an excellent ring of equal characteristic. The paper under review studies several complexes associated to the normalized dualized complex of \(R\), when \(R\) is \(F\)-injective in prime characteristic or when \(R\) defines a Du Bois singularity in characteristic zero. Specifically the authors assume that either (1) \(R\) is F-injective of prime characteristic or (2) \(R\) is essentially of finite type over a field of characteristic zero and is Du Bois. Now fix an integer \(j>0\) and assume that the local cohomology modules \(H^i_{\mathfrak{m}}(R)\) have finite length for all \(0\leq i<j\). The main result of the paper shows that, under these conditions, the trucation of the normalized dualizing complex \(\tau_{>-j}\omega_R^\cdot\) is quasi-isomorphic to a complex of \(k\)-vector spaces. This theorem recovers a result of Ma that asserts that \(F\)-injective singularities with isolated non-Cohen-Macaulay locus are [\textit{L. Ma}, Math. Ann. 362, No. 1--2, 25--42 (2015; Zbl 1398.13003)]. Additionally, it answers a question of Schwede and Takagi, by showing that Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum, generalizing work of \textit{M. N. Ishida} [in: Algebraic and topological theories. Papers from the symposium dedicated to the memory of Dr. Takehiko Miyata held in Kinosaki, October 30- November 9, 1984. Tokyo: Kinokuniya Company Ltd.. 387--390 (1986; Zbl 0800.32007)]. Moreover, the paper contains a more elementary approach of the main result in the case where the characteristic is prime and the field \(k\) is perfect. It also contains generalizations of the main result by involving some additional complexes related to normalized dualizing complex \(\omega_R^{\cdot}\). \(F\)-injective; Du Bois; dualizing complex; local cohomology Multiplier ideals, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry The dualizing complex of \(F\)-injective and Du Bois singularities
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((G_{{\mathbb{R}}},\rho,V_{{\mathbb{R}}})\) be a regular prehomogeneous vector space. Suppose that \(G_{{\mathbb{R}}}\) is reductive and the singular set S is an irreducible hypersurface defined by the irreducible polynomial P(x) of degree d. Let \(V_ 1\cup V_ 2\cup...\cup V_{\ell}=V_{{\mathbb{R}}}-S_{{\mathbb{R}}}\) be the connected component decomposition. Consider the complex powers \(| P(x)|^ s|_{V_ i}\) \((i=1,2,...,\ell)\), which define distributions on \(V_{{\mathbb{R}}}\) with a meromorphic parameter \(s\in {\mathbb{C}}\). In this paper, the author proves that there exist \(d\ell /2\) linear relations among the negative Laurent coefficients of the distributions \(| P(x)|^ s|_{V_ i}\) \((i=1,2,...,\ell)\) with respect to s. The author uses these relations to prove that the image of the map M: \({\mathcal S}(V_{{\mathbb{R}}})\to C^{\infty}(R_{>0},{\mathbb{C}}^{\ell})\) where \(M_ f(t)=(M_{i,f}(t))_{1\leq i\leq \ell}\) is the integral of the test function \(f\in {\mathcal S}(V_{{\mathbb{R}}})\) along each component of the fibre \(P^{-1}(t)\), is a subspace of an \({\mathcal S}({\mathbb{R}})\)-module generated by \(d\ell /2\) elements which he determines. invariant distributions; regular prehomogeneous vector space; connected component; linear relations; Laurent coefficients General properties and structure of real Lie groups, Linear algebraic groups over the reals, the complexes, the quaternions, Homogeneous spaces and generalizations Relations entre les fonctions moyennes sur un espace préhomogène. (Relations between mean functions on a prehomogeneous space)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Standard Monomial Theory deals with the problem of producing a basis of \(H^0(X, L)\) for a Schubert variety \(X\) inside a homogeneous space \(G/P\) and a line bundle \(L\) on \(G/P\) that can be understood by some combinatorial objects. One particularly desired property of such a basis is compatibility with special subvarieties. For example, a standard monomial basis of \(H^0(G/P, L)\) should restrict to a basis of \(H^0(X, L)\).
The Bott-Samelson variety is a classic desingularization of a Schubert variety. The intersection of a Schubert variety with an opposite Schubert variety is called a Richardson variety. The author considers desingularizations \(\Gamma\) of these Richardson varieties in the case of a complete flag variety of type \(A_n\) as a fibre of a projection from a certain Bott-Samelson variety \(Z\). The article deals with a version of Standard Monomial Theory on these desingularizations.
In case of the Bott-Samelson variety, \textit{V. Lakshmibai}, \textit{P. Littelmann} and \textit{P. Magyar} [in: Representation theories and algebraic geometry. Proceedings of the NATO Advanced Study Institute, Montreal, Canada, July 28--August 8, 1997. Broer, A. (ed.) et al., Representation theories and algebraic geometry. Proceedings of the NATO Advanced Study Institute, Montreal, Canada, July 28--August 8, 1997. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 514, 319--364 (1998; Zbl 1013.17014)] had previously defined a family of globally generated line bundles \(L_m\) and given a basis for \(H^0(Z, L_m)\) by objects called standard tableaux. The author deals with the question whether these standard monomial bases are compatible with all choices of \(\Gamma\), i.e. whether they restrict to a basis of \(H^0(\Gamma, L_m)\). The main result is that this is true if \(L_m\) is very ample. Moreover, there is a concrete criterion for which basis elements become zero after restricting and have to be removed. Richardson varieties; standard monomial theory; flag variety Grassmannians, Schubert varieties, flag manifolds, Affine algebraic groups, hyperalgebra constructions Standard monomial theory for desingularized Richardson varieties in the flag variety \(\mathrm{GL}(n)/B\)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\) be a commutative noetherian ring and let \(I\) be an ideal of \(R[ x_1, \dots, x_n]= R[ x]\). The morphism \(\psi: R\mapsto R[ x]/I\) defines a family of algebraic varieties as follows: Let \(p\) be a prime ideal of \(R\) (or an element of \(\text{Spec } R\)) and let \(K(p)\) be the quotient field of the localization \(R_p\) of \(R\) at \(p\), then we have an algebraic variety in \(A^n_{K (p)}\) defined by \(K(p) [x]/ I(p)\) where \(I(p)= I.K (p) [x]\). When \(p\) varies, these varieties are called fibers of \(\psi\). On the other hand, when \(\psi\) is flat, many properties are preserved in the fibers. The main objective of this paper is to characterize flatness of \(\psi\) by studying the relationship with the notions of Gröbner and standard bases. When \(R\) is principal, we obtain an algorithm to compute the maximal generic open set of flatness of \(\text{Spec } R\) and then we give some applications related to this situation. Gröbner bases; polynomial ideal; noetherian ring; standard bases; open set of flatness of Spec Assi, A.: On flatness of generic projections. J. symbolic comput. 18, No. 5, 447-462 (1994) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Projective and free modules and ideals in commutative rings, Relevant commutative algebra, Polynomial rings and ideals; rings of integer-valued polynomials On flatness of generic projections
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Hereafter, we refer to [\textit{R. Hartshorne}, Algebraic geometry. York-Heidelberg-Berlin: Springer-Verlag (1977; Zbl 0367.14001)] for any unexplained terminology.
Let \(\pi: Y\rightarrow X\) be a morphism of reduced schemes of finite type over a field \(\mathbb{K}\) and suppose that \(\mathfrak{a}\) is a sheaf of ideals on \(X\). We say that \(\pi\) is a log resolution of \(\mathfrak{a}\) provided it satisfies the following four conditions. {\parindent=6mm \begin{itemize}\item[1.] \(\pi\) is birational and proper. \item[2.] \(Y\) is smooth over \(\mathbb{K}\). \item[3.] \(\mathfrak{a}\mathcal{O}_Y =\mathcal{O}_Y (-G)\) is an invertible sheaf corresponding to a divisor, namely \(-G\). \item[4.] If \(E\) is the exceptional set of \(\pi\), then \(\operatorname{Supp} (G)\cup E\) has simple normal crossings.
\end{itemize}} Moreover, we say that \(\pi\) is a strong log resolution if it is a log resolution and \(\pi\) is an isomorphism outside of the subscheme \(V(\mathfrak{a})\) defined by \(\mathfrak{a}\). It is well known, by the celebrated results obtained by \textit{H. Hironaka} in [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)], that log resolutions always exist if the characteristic of \(\mathbb{K}\) is zero and strong log resolutions always exist in case \(X\) is smooth and \(\mathbb{K}\) has characteristic zero. However, despite the effort of many researchers the existence of resolution of singularities over a field of prime characteristic is still an open question in general.
Loosely speaking, in characteristic zero Hironaka showed that it is possible to obtain a (not necessarily unique) resolution of singularities through a sequence of normalizations and blowups at smooth centers. It turns out that this strategy does not work in prime characteristic; regardless, it is natural to ask the following:
Question. Is there a notion in prime characteristic which might play the role of blowup in characteristic zero?
Inspired by this question, \textit{T. Yasuda} in [Am. J. Math. 134, No. 2, 349--378 (2012; Zbl 1251.14002)] defined the so-called \(e\)th \(F\)-blowup, with the hope that it might play the same role as the blowup in characteristic zero. Roughly speaking, the idea is, given a (possibly singular) algebraic variety over a field of prime characteristic, to construct another algebraic variety \(Y\) on which the Frobenius endomorphism is flat. The idea is, of course, inspired by the classical result obtained by \textit{E. Kunz} in [Am. J. Math. 91, 772--784 (1969; Zbl 0188.33702)] which characterizes the regularity of a commutative reduced ring \(R\) of prime characteristic in terms of the fact that the Frobenius map on \(R\) is flat.
Yasuda's definition works as follows. Let \(\mathbb{K}\) be an algebraically closed field of prime characteristic, and let \(X\) be an algebraic variety over \(\mathbb{K}\) of dimension \(n\). The \(e\)th \(F\)-blowup \(\operatorname{FB}_e (X)\) of \(X\) is defined to be the closure of the subset
\[
\{ (F^e)^{-1} (x)\mid x\in X\text{ smooth}\}\subseteq\operatorname{Hilb}_{p^{ne}} (X^{(e)}),
\]
where \(X^{(e)}\) is the ringed space \((X,F_*^e \mathcal{O}_X)\) and \(\operatorname{Hilb}_{p^{ne}} (X^{(e)})\) denotes the Hilbert scheme of zero dimensional subschemes of \(X^{(e)}\) of length \(p^{ne}\). We refer to [Zbl 0188.33702] for more details.
From now on, we restrict our attention to the case of surfaces. In this case, given a surface \(S\) it is known that, in any characteristic, there exists a minimal resolution of singularities \(S_0\rightarrow S\). So, it is natural to ask the following:
Question. Let \(\mathbb{K}\) be an algebraically closed field of prime characteristic, let \((S,x)\) be a normal surface singularity and let \(S_0\rightarrow S\) be the minimal resolution. When is \(\operatorname{FB}_e (S)\) equal to the minimal resolution \(S_0\)?
It is true that \(\operatorname{FB}_e (S)=S_0\) for \(e\gg 0\) if either \(S\) is a toric singularity, a tame quotient singularity, or an \(F\)-regular double point.
We wish to introduce an additional notion before starting properly our review. Let \(R\) be an integral domain of prime characteristic \(p\) which is \(F\)-finite. We say that \(R\) is strongly \(F\)-regular if, for any nonzero element \(c\in R\), there exists a power \(q=p^e\) such that the inclusion map \(c^{1/q}R\hookrightarrow R^{1/q}\) splits as an \(R\)-module homomorphism.
In the paper under review, the author shows as main result that if \((S,x)\) is an strongly \(F\)-regular surface singularity, then its \(e\)th \(F\)-blowup \(\operatorname{FB}_e (S)\) coincides with the minimal resolution of \(X\) for \(e\gg 0\). This result generalizes an earlier one obtained by \textit{N. Hara} and \textit{T. Sawada} in [RIMS Kôkyûroku Bessatsu B24, 121--141 (2011; Zbl 1228.13009)] which was proved for \(F\)-rational double points. One of the technical tools developed by the author for that purpose, which is interesting in its own right, is a nice characterization of complete strongly \(F\)-regular rings of dimension \(2\) over \(\mathbb{K}\) (cf. Theorem 2.1). Finally, the author raises several questions related with these topics in order to stimulate further research about \(F\)-blowups and strongly \(F\)-regular rings. Hara, Nobuo: F-blowups of F-regular surface singularities, Proc. amer. Math. soc. 140, 2215-2226 (2012) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics F-blowups of F-regular surface singularities
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system It has been proved by \textit{J. Alexander} [Compos. Math. 68, No. 3, 305- 354 (1988; Zbl 0675.14025)] that there exists a hypercubic in \(\mathbb{P}^ 4\) which is singular at seven general given points, in contrast with a counting of dimensions. The authors show that this hypercubic is singular along a curve and hence conclude that it is in fact the secant variety to the unique rational normal quartic passing through the given seven points.
For this, they prove the following general fact in characteristic zero: Let \(L\) be a \(d\)-dimensional linear system on a smooth \(r\)-dimensional variety. Assume that the subsystem \(L'\) of divisors which are singular at \(n\) general given points has dimension bigger than the expected \(d- n(r+1)\). Then, for each divisor \(D\) in \(L'\), at least one of the \(n\) given points is not isolated in the singular locus of \(D\). singularities of linear systems; dimension of linear systems; hypercubic; divisors C. Ciliberto and A. Hirschowitz: ''Hypercubique de \(\mathbb{P}\)4 avec sept points singulieres génériques'', C. R. Acad. Sci. Paris, Vol. 313(1), (1991), pp. 135--137. Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry, Projective techniques in algebraic geometry, \(3\)-folds Hypercubiques de \(\mathbb{P}^ 4\) avec sept points singuliers génériques. (Hypercubics in \(\mathbb{P}^ 4\) with seven generic singular points)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system From the preliminaries: ``Let \(P_ 1,...,P_ s\in {\mathbb{P}}^ n_ k\) denote s distinct points in projective n-space \({\mathbb{P}}^ n_ k\). Throughout this paper, k will denote an algebraically closed field whose characteristic is arbitrary (except for a couple of examples near the end). Let \(S=k[X_ 0,...,X_ n]\) denote the polynomial ring in \(n+1\) variables \(X_ 0,...,X_ n\) over k. If I denotes the ideal of \(P_ 1,...,P_ s\) in S, then I is a perfect, unmixed, radical ideal of grade n. Let \(R=S/I\), the coordinate ring of \(P_ 1,...,P_ s\). R is a standard graded k-algebra, Cohen-Macaulay of dimension one and has projective dimension n as an S-module. Thus, R has a minimal, free resolution \(\Gamma\) of the form:
\[
(1)\;\Gamma:0\to \oplus^{\beta_ n}_{i=1}S(-d_ i^{(n)})\to^{\phi_ n}...\to \oplus^{\beta_ 2}_{i=1}S(-d_ i^{(2)}\to^{\phi_ 2}\to \oplus^{\beta_ 1}_{i=1}S(-d_ i^{(1)})\to^{\phi_ 1}S\to R\to 0.
\]
In (1), \(\beta_ 1,...,\beta_ n\) are the nontrivial Betti numbers of R. Each \(\phi_ i\) is a homogeneous S-module homomorphism of degree zero. Consequently, \(\phi_ i\) can be represented by a \(\beta_ i\times \beta_{i-1}\) matrix \((\alpha_{pq}^{(i)})\) where \(\alpha_{pq}^{(i)}\) is a homogeneous form in S of degree \(\partial (\alpha_{pq}^{(i)})=d_ p^{(i)}-d_ q^{(i-1)}\). \(\Gamma\) being minimal means every \(\alpha^{(i)}_{p,q}\in (X_ 0,...,X_ n)\). The \(d_ i^{(i)}\) in (1) are called the twisting numbers of R and, along with \(\beta_ 1,...,\beta_ n\), are unique. We say \(P_ 1,...,P_ s\) have a pure resolution of type \((d_ 1,...,d_ n)\) if, in the minimal resolution (1) of R, we have for all \(j=1,...,n\) and for all \(i=1,...,\beta_ j\), \(d_ i^{(j)}=d_ j\). Thus, \(P_ 1,...,P_ s\) have a pure resolution of type \((d_ 1,...,d_ n)\) with Betti numbers \(\beta_ 1,...,\beta_ n\) if and only if the minimal free resolution \(\Gamma\) of R has the simple form:
\[
(2)\;\Gamma:0\to S(-d_ n)^{\beta_ n}\to...\to S(-d_ 2)^{\beta_ 2}\to S(-d_ 1)^{\beta_ 1}\to S\to RK0.
\]
We note that the minimality of \(\Gamma\) implies \(0<d_ 1<d_ 2<...<d_ n\) in (2). If \(\Gamma\) is a pure resolution of type \((e,e+m,...,e+(n-1)m)\), we shall abbreviate our notation and say \(\Gamma\) is a pure resolution of type \(<e;m>\). Thus, when \(m=1\), a pure resolution of type \(<e;1>\) is just the usual notion of a linear resolution. Finally, we say \(\Gamma\) is almost linear if \(\Gamma\) is a pure resolution of type \((e,e+m,...,e+(n-2)m,d_ n).\)
In this paper, we investigate what points \(P_ 1,...,P_ s\in {\mathbb{P}}^ n_ k\) have pure resolutions. This problem is almost hopeless unless we put more conditions on the \(P_ i\). One such condition which readily comes to mind is to control the Hilbert function \(H_ R(t)\) of the \(P_ i\). If A is any standard graded k-algebra, we shall let \(A_ t\) denote the t-th homogeneous piece of A. The Hilbert function, \(H_ A(t)\), of A is then given by \(H_ A(t)=\dim_ k\{A_ t\}\). For example, \(H_ S(t)\) is equal to the binomial coefficient \(\binom{n+t}{n}\) for all \(t\geq 0\). The Poincaré series, \(F_ A(z)\), of A is the formal power series \(\sum^{\infty}_{t=0}H_ A(t)z^ t\). Set \(\nu(t)=\binom{n+t}{n}\). We say s distinct points \(P_ 1,...,P_ s\in {\mathbb{P}}^ n_ k\) are in generic s-position if \(H_ R(t)=\min \{s,\nu (t)\}\) for all \(t\geq 0\). We say \(P_ 1,...,P_ s\in {\mathbb{P}}^ n_ k\) are in uniform position if for every \(t=1,...,s\), and for every subset \(P_{i_ 1},...,P_{i_ t}\) (of \(P_ 1,...,P_ s)\) consisting of t distinct points, we have \(P_{i_ 1},...,P_{i_ t}\) are in generic t-position. Most sets of s points in \({\mathbb{P}}^ n_ k\) are in generic (uniform) position in the sense that the points in generic s-position (uniform position) in \({\mathbb{P}}^ n_ k\) form a dense open subset of \({\mathbb{P}}^ n_ k\times...\times {\mathbb{P}}^ n_ k\) (s times).
In this note, we investigate what points in generic position have pure resolutions and what these resolutions look like.'' minimal free resolution of the coordinae ring of s distinct points. points in generic position; pure resolutions Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Cycles and subschemes A note on pure resolutions of points in generic position in \({\mathbb{P}}^ n_ k\)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A theory of the resolution of singularities should not only prove that every space \(Z\) can be resolved by a modification \(Z' \to Z\), but the resolution should be
\begin{itemize}
\item canonical in the sense that the theory distinguishes one resolution \(Z_{res} \to Z\);
\item constructive in the sense that there is an explicit algorithmic procedure to obtain the distinguished resolution \(Z_{res} \to Z\);
\item functorial in the sense that for every smooth morphism \(Y \to Z\), the resolution \(Y_{res} \to Y\) is the pull-back (in the appropriate category) of \(Z_{res} \to Z\).
\end{itemize}
The article is the first in an announced series which establishes such a resolution for morphisms \(Z \to B\) of fine saturated logarithmic Deligne-Mumford stacks. The article itself deals only with the case \(B = \mathrm{Spec}(k)\) the trivial log point, i.e., with the absolute case.
The starting point is resolution of logarithmic schemes. The authors ask for an extended functoriality principle, i.e., the resolution should not only be functorial for classically smooth maps \(Y \to Z\) but also for log smooth maps \(Y \to Z\). This makes it necessary to resolve via blowups of Kummer centers, i.e., ideals in the Kummer étale topology of \(Z\). Such a blowup is not necessarily a logarithmic scheme but a logarithmic Deligne-Mumford stack; thus the natural setup for the resolution are (fine saturated) logarithmic Deligne-Mumford stacks. A log smooth Deligne-Mumford stack is called a toroidal orbifold.
The algorithm comes in two versions, an embedded one -- the principalization of ideals on a toroidal orbifold -- and a non-embedded one -- the resolution of singularities.
The algorithm does not specialize to a classical one for trivial logarithmic structures. When it starts with a variety \(Z\) with trivial logarithmic structure, then \(Z_{res}\), in general, neither has trivial logarithmic structure nor is a scheme but a honest toroidal orbifold.
The algorithm is less complicated than algorithms for resolving singularities of classical schemes since it does not need to take separate care about the exceptional divisor, but it is nonetheless intricate. It employs a logarithmic version of the induction over hypersurfaces of maximal contact to reduce the log order of marked ideals. Additionally, in several so-called cleaning processes it must be ensured that the ideal to resolve has a nice interplay with the logarithmic structure.
After this article was finished, Ming Hao Quek -- a student of one of the authors -- has found a simpler, more canonical, and presumably faster algorithm to resolve logarithmic singularities in the same setup, see [\textit{M.~H.~Quek}, ``Logarithmic resolution via weighted toroidal blow-ups'', Preprint, \url{arXiv:2005.05939}]. This is achieved by allowing more general centers which are no longer ideals in the Kummer étale topology; it is a logarithmic variant of the \emph{dream algorithm} of [\textit{D.~Abramovich} et al., ``Functorial embedded resolution via weighted blowings up'', Preprint, \url{arXiv:1906.07106}]. resolution of singularities; logarithmic geometry; algebraic stacks Global theory and resolution of singularities (algebro-geometric aspects), Generalizations (algebraic spaces, stacks), Logarithmic algebraic geometry, log schemes Principalization of ideals on toroidal orbifolds
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Take a linear system of plane projective curves \(C(\tau) (\tau \in {\mathbb C})\) spanned by two irreducible curves \(C\) and \(C'\) of degree \(d\).
Choose a generic line at infinity \(L_{\infty}: Z=0\) such that the affine space \({\mathbb C}^2:= {\mathbb P}^2 - L_{\infty}\) contains all the base points \(C \cap C'\). In this affine space \(C(\tau)\) is defined by \(\tau f(x,y)+(1- \tau)g(x,y) =0\).
Consider a union of generic \(r\) curves in this pencil : \(C(\overrightarrow{\tau}):=C({\tau}_1) \cup \dots C({\tau}_r)\), where the parameters \({\tau}_1, \dots , {\tau}_r\) are generic so that the topology of the pair \(({\mathbb P}^2,C(\overrightarrow{\tau}))\) does not depend on the choice of \({\tau}_1, \dots, {\tau}_r\). The pencil \(C(\tau)\) is called abelian if \(\pi_1({\mathbb P}^2-C(\tau_1) \cup C(\tau_2))\) is abelian for any generic \(\tau_1, \tau_2\).
Let \(\phi: {\mathbb C}^2 \rightarrow {\mathbb C}^2\) the \(d^2\)-fold branched covering, where \(\phi(x,y)=(f(x,y),g(x,y))\) and put \(L(\overrightarrow{\tau})= L({\tau}_1) \cup \dots \cup L({\tau}_r)\) , where \(L(\tau)\) is defined by \(\tau x+(1-\tau)y=0\). Then \(\phi^{-1}(L(\overrightarrow{\tau}))=C(\overrightarrow{\tau})\).
Main result of this paper is that \(\phi\) induces an isomorphism \(\phi_{\natural} : \pi_1({\mathbb C}^2-C((\overrightarrow{\tau})) \rightarrow \pi_1({\mathbb C}^2-L((\overrightarrow{\tau}))\) .
In particular \(\pi_1({\mathbb C}^2-C(\overrightarrow{\tau}))\) is isomorphic to \({\mathbb Z}\times F(r-1)\), \(C(\overrightarrow{\tau})\) is a curve of non-torus type and the Alexander polynomial of \(C(\overrightarrow{\tau} )\) is given by \((t^r-1)^{r-2}(t-1)\) (see [in: Singularities II. Geometric and topological aspects. Proceedings of the international conference ``School and workshop on the geometry and topology of singularities'' in honor of the 60th birthday of Lê Dũng Tráng, Cuernavaca, Mexico, January 8--26, 2007. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 475, 151--167 (2008; Zbl 1185.14024)] for an analogous result by the author, for smooth pencils of curves of a strict non-torus type). generic pencil curves; abelian pencil of curves Special algebraic curves and curves of low genus, Coverings of curves, fundamental group Topology of abelian pencils of curves
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \({\mathcal F}\) be the flag manifold of an \(n\)-dimensional vector space \(V\). To each unipotent transformation \(u\) of \(V\), denote by \({\mathcal F}_ u\) the variety of flags fixed by \(u\). In this paper it is shown that the element of \(S_ n\) which determines the relative position of two irreducible components is equal to the element given by the Robinson-Schensted correspondence applied to the associated two standard tableaux.
As pointed out by the author a proof of this result has also been given by \textit{N. Spaltenstein} [in: Classes unipotentes et sous groupes de Borel. Lect. Notes Math. 946. Berlin etc.: Springer Verlag (1982; Zbl 0486.20025)] although the Robinson-Schensted process is there only implicit. flag manifold; unipotent transformation; variety of flags; irreducible components; Robinson-Schensted correspondence; standard tableaux Steinberg, Robert, An occurrence of the Robinson-Schensted correspondence, J. Algebra, 113, 2, 523-528, (1988) Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Representations of finite symmetric groups, Combinatorial aspects of representation theory, Multiply transitive finite groups An occurrence of the Robinson-Schensted correspondence
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We study the commutative algebras \(Z_{JK}\) appearing in \textit{K. A. Brown} and \textit{K. R. Goodearl}'s [Math. Sci. Res. Inst. Publ. 68, 63--91 (2015; Zbl 1359.16035)] extension of the \(\mathcal {H}\)-stratification framework, and show that if \(A\) is the single parameter quantized coordinate ring of \(M_{m,n}\), \(\operatorname{GL}_{n}\) or \(\operatorname{SL}_{n}\), then the algebras \(Z_{JK}\) can always be constructed in terms of centres of localizations. The main purpose of the \(Z_{JK}\) is to study the structure of the topological space \(\operatorname{spec}(A)\), which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real \( m \times n\) matrices in terms of its restricted permutation. quantum matrices; torus-invariant prime ideals; totally nonnegative cells; Grassmann necklaces Ring-theoretic aspects of quantum groups, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Quantum groups (quantized function algebras) and their representations From Grassmann necklaces to restricted permutations and back again
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R, m)\) be a complete commutative Noetherian local ring, \(I\) an ideal of \(R, M\) an \(R\)-module (not necessarily \(I\)-torsion) and \(N\) a finitely generated Rmodule with Supp\(_R(N)\subseteq V(I)\). It is shown that if \(M\) is \(I\)-ETH-cominimax (i.e. Ext\(^i_R(R/I, M)\) is minimax (or Matlis reflexive), for all \(i\geq 0\)) and \(\dim M\leq 1\) or more generally \(M\in FD_{\leq 1}\), then the \(R\)-module Ext\(^n_ R(M, N)\) is finitely generated, for all \(n\geq 0\). As an application to local cohomology, let \(\Phi\) be a system of ideals of \(R\) and \(I\in\Phi\), if \(\dim M/aM\leq 1\) (e.g., \(\dim R/a\leq 1\)) for all \(a\in\Phi\), then the \(R\)-modules Ext\(^j_ R(H^i_{\Phi}(M), N)\) are finitely generated, for all \(i\geq 0\) and \(j\geq 0\). Similar results are true for local cohomology defined by a pair of ideals and ordinary local cohomology modules. cominimax modules; local cohomology; Matlis reflexive modules; minimax modules Local cohomology and commutative rings, Commutative Noetherian rings and modules, Local cohomology and algebraic geometry Finiteness properties of extension functors of cominimax modules
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The sheaf of principal parts \(J^k(E)\) has been studied by many authors (DiRocco, Grothendieck, Laksov, Maakestad, Perkinson, Piene, Sommese etc). The sheaf \(J^k(E)\) of an \(O_X\)-module \(E\) where \(X\) is a scheme has a left and right structure as \(O_X\)-module and the left structure of \(J^k(E)\) has been studied by several authors in the case where \(X\) is projective \(n\)-space over an algebraically closed field of characteristic zero. The aim of the paper under review is to complete this study and to relate it to the theory of representations of quivers.
The projective space \(\mathbb{P}(V^*)\) may be realized as a quotient \(\mathrm{SL}(V)/P\) where \(P\) is a parabolic subgroup of \(\mathrm{SL}(V)\) and there is an equivalence of categories between the category of \(P\)-modules and the category of \(\mathrm{SL}(V)\)-linearized vector bundles on \(\mathbb{P}=\mathbb{P}(V^*)\). The category \(C\) of vector bundles on \(P\) with an \(\mathrm{SL}(V)\)-linearization is an abelian category, hence by Freyd's full embedding theorem it follows the category \(C\) is equivalent to a full subcategory of the category \(mod(A)\) of left modules on an associative ring \(A\). One aim of the paper is to describe the associative ring \(A\) associated to projective space \(P\) and to construct the \(A\)-module corresponding to the sheaf of principal parts \(J^k(E)\).
In section one of the paper the author gives the motivation for the writing of the paper and the main results of the paper.
In section two of the paper the author gives the definition of the sheaf of principal parts \(J^k(E)\) of any \(O_X\)-module \(E\) on any scheme \(X\) following the standard construction using the infinitesimal neighborhood of the diagonal and mentions some properties of this construction: He gives a description of the fiber of the principal parts, the fundamental exact sequences and the relationship with sheaves of differential operators.
In section three the author introduce the concepts of algebraic groups, homogeneous spaces and homogeneous vector bundles and mention the fact that if \(E\) is a \(G\)-linearized homogeneous vector bundle on a homogeneous space \(G/H\) it follows \(J^*(E)\) has a canonical \(G\)-linearization. The author mentions the notions of a Cartan decomposition of a parabolic Lie algebra, the notion of a maximal weight vector of a \(p\)-module where \(p\) is a parabolic Lie algebra and the notion of an irreducible homogeneous vector bundle. The author ends section three with an introduction to the notion of quiver representations. He also constructs the quiver \(Q_V\) associated to projective space \(\mathbb{P}=\mathbb{P}(V^*)\). He moreover gives an explicit construction of the equivalence between the category of representations of \(Q_V\) and the category of homogeneous vector bundles on \(P\).
In section four the author mentions known results on \(J^k(O(d))\) on \(P\) and introduces some notions defined in [\textit{D. Perkinson}, Compos. Math. 104, 27--39 (1996; Zbl 0895.14016)]. He uses these notions and some explicit formulas to prove the existence of a decomposition \(J^k(O(d))\cong Q_{k,d}\oplus J^d(O(d))\) where \(Q_{k,d}\) is an explicitly defined vector bundles on \(P\). The author defines a map
\[
n^{k-d}:S^k(V)\otimes O(d-k) \rightarrow S^k(V) \otimes O_P
\]
and an isomorphism \(Q_{k,d}\cong \ker(n^{k-d})\). He proves that the map \(n^{k-d}\) is an \(\mathrm{SL}(V)\)-invariant differential operator. The author moreover proves some properties of the bundles \(Q_{k,d}\) and use these properties to give an explicit construction of the \(Q_V\)-representation of \(J^k(O(d))\). The paper ends with a proof of the fact that the Taylor truncation map has maximal rank in the cases where \(h \leq k\).
Note: In Proposition 4.6 the author states the existence of an \(\mathrm{SL}(V)\)-equivariant isomorphism
\[
J^k(O(d)) \cong S^k(V) \otimes O(d-k)
\]
In other papers [the reviewer, Proc. Am. Math. Soc. 133, No. 2, 349--355 (2005; Zbl 1061.14040)] it was proved that \(J^k(O(d))\cong S^k(V^*)\otimes O(d-k)\). One may suspect there is an error in the paper since \(S^k(V)\) and \(S(V^*)\) are different as \(\mathrm{SL}(V)\)-modules. The difference between the author's paper and the reviewer's paper is that Re is considering \(\mathbb{P}(V)\) -- projective space parametrizing lines in \(V^* \) where Maakestad is considering \(\mathbb{P}(V^*)\) -- projective space parametrizing lines in \(V\). principal parts; quiver representation; stable; vector bundle; projective space Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results Principal parts bundles on projective spaces and quiver representations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(A\) be a two-dimensional complete regular local ring: A one-dimensional reduced complete Noetherian local ring \(R=A/I\) is called plane curve singularity; a branch is a height one prime ideal \(p\) of \(A\); if \(p\) and \(q\) are branches, denote by \(\mu (p,q)\) the intersection multiplicity \(l_A(A/(p+q))\) of \(p\) and \(q\). An equivalence relation \(\approx\) on the set of branches of \(A\) is said to be \(\mu\)-compatible if it has the following property: Let \(\{p_1, \cdots , p_r \}\) and \(\{q_1, \cdots, q_r \}\) be sets of branches in \(A\) such that \(p_i \approx q_i\) and \(\mu(p_i,p_j)= \mu(q_i,q_j)\) for all \(i\) and \(j\); then for every branch \(p\) there is a branch \(q \approx p\) such that \(\mu (p,p_j)= \mu(q,q_j)\) for all \(j\). The notion of (a)-equivalence of plane curve singularities introduced by Zariski has been extended by Granja and Sanchez-Giralda to branches of any ring \(A\) as above.
The main purpose of this paper is to give an axiomatic description of (a)-equivalence of branches in \(A\) provided that the residue class field \(k\) of \(A\) satisfies the condition (d): if \(k \subseteq k_0 \subseteq k_1 \subseteq \cdots \subseteq k_s\) is a tower of simple algebraic extensions of \(k\), the set \(\{[l:k_s] |k_s \subseteq l\) simple algebraic\} depends only on the degree sequence \(\{[k_i:k_{i-1}]\mid i=1, \dots , s \}\);
in fact theorem 1 says that if \(k\) satisfies condition (d), (a)-equivalence is the coarsest \(\mu\)-compatible equivalence relation \(\sim\) on the set of branches in \(A\), i.e., if \(\approx\) is any other relation, then \(p \approx q\) implies \(p \sim q\). The author gives also some application of this result. equivalent branches of plane curves Singularities of curves, local rings On the equivalence of plane curve singularities
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A regular neighborhood of a linear chain of \(2\)-spheres is a smooth \(4\)-manifold obtained by taking a finite sequence of disk bundles over the \(2\)-sphere with prescribed Euler characteristic and plumbing each to the next.
Let \(1\leq q<p\) and let \(p\) and \(q\) be coprime. Let \(B_{p,q}\) be a smooth, rational homology \(4\)-ball with boundary the lens space \(L(p^2,pq-1)\).
Using techniques from the minimal model program for \(3\)-dimensional complex algebraic varieties, the authors prove that \(B_{p,q}\) can be smoothly embedded in a regular neighborhood of a certain linear chain of \(2\)-spheres. flip; rational blow-down/blow-up; rational homology ball Embeddings in differential topology, Differentiable structures in differential topology, Deformations of singularities, Topology of Euclidean 4-space, 4-manifolds Smoothly embedded rational homology balls
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(A\) be the homogeneous coordinate ring of a Grassmannian over \(K\) (of characteristic zero). The Poincaré resolution of the graded \(K\)-algebra \(A\) is the minimal \(A\)-free resolution of \(K\) (e.g. for \(\mathbb{P}_ n\), this is the Koszul complex corresponding to \(n+1\) general hyperplanes). The authors show, in a constructive way, that this resolution is linear (more concretely, the \(m\)-th term of it is concentrated in degree \(m)\). They also prove this property for the minimal \(A\)-free resolution of the tautological bundle. homogeneous coordinate ring of a Grassmannian; Poincaré resolution; Koszul complex Grassmannians, Schubert varieties, flag manifolds, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) A note on the Poincaré resolution of the coordinate ring of the Grassmannian
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system If M is a graded module over a polynomial ring with all generators in the same degree, say 0, then the linear part of the resolution of M consists of the syzygies of degree 1, the syzygies on these of degree 2, etc. Linear parts of resolutions are far easier to study than the resolutions themselves, and there are interesting criteria, related to the classical notions of ''Castelnuovo regularity'' in algebraic geometry, for a resolution to be equal to its linear part. The first goal of the paper under review is to treat these matters.
Probably the most familiar example of an ideal with linear resolution is that of the \(p\times p\) minors of a \(p\times q\) matrix with linear entries, in the case where the \(p\times p\) minors have generic depth, \(q- p+1\). In the case \(p=2\) these examples appear in geometry as rings of minimal multiplicity (or projective varieties of minimal degree). In fact the classification theorem of Bertini implies that varieties of minimal degree all are Cohen-Macaulay, and their ideals linear resolutions. The second purpose of this article is to give a direct, ring-theoretic treatment of the connection between minimal degree, and these other, homological notions. local cohomology; free resolution; syzygies; graded module over polynomial ring; Bertini classification; Linear parts of resolutions; Castelnuovo regularity; rings of minimal multiplicity; projective varieties of minimal degree \beginbarticle \bauthor\binitsD. \bsnmEisenbud and \bauthor\binitsS. \bsnmGoto, \batitleLinear free resolutions and minimal multiplicity, \bjtitleJ. Algebra \bvolume88 (\byear1984), page 89-\blpage133. \endbarticle \OrigBibText David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity . J. Algebra 88 (1984), 89-133. \endOrigBibText \bptokstructpyb \endbibitem Multiplicity theory and related topics, Determinantal varieties, Complexes, Projective and free modules and ideals in commutative rings, Local cohomology and algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Global theory and resolution of singularities (algebro-geometric aspects), (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Linear free resolutions and minimal multiplicity
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A reduced hypersurface \(D\subset \mathbb{C}^n=V\) is a linear free divisor if the sheaf \(\mathrm{Der}(-\log\,D)\) of logarithmic vector fields along \(D\) is freely generated by the germs of globally defined linear vector fields. By \textit{K. Saito}'s criterion (see [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265--291 (1980; Zbl 0496.32007)], Theorem 1.8), such a \(D\) is the zero locus of a reduced homogeneous polynomial \(f\) of degree \(n\). In this sense, this class of divisors is opposite to the class of central hyperplane arrangements, where the irreducible components are linear but the basis of logarithmic vector fields is in general not. (Only the normal crossing divisor is a member of both classes).
For a general \(D\subset V\), let \(G\subset \mathrm{GL}(V)\) be the largest connected subgroup preserving \(D\). An unpublished result of Brion, states that \(D\) is a linear free divisor if and only if: 1) The complement \(V\setminus D\) is an open \(G\) orbit and the corresponding isotropy groups are finite. 2) The smooth part in \(D\) of each irreducible component of \(D\) is a unique \(G\)-orbit, and the corresponding isotropy groups are extensions of finite groups by the multiplicative group \(\mathbb C^*\). In particular, \(V\) is a prehomogeneous vector space (i.e. it contains an open \(G\) orbit). A proof of such a result is given by the authors.
The author are interested to the following two questions: 1) Classify the linear free divisors. 2) Classify the linear free divisors which satisfy the logarithmic comparison theorem. Moreover, there are especially interested to the divisors in prehomogeneous vector spaces which are quiver representations.
Let \(D\subset V\), \(f\) and \(G\) as before and let \(H\) be the identity component of the stabilizer of \(f\) in \(G\). A linear free divisor \(D\) is called reductive (resp. abelian) if \(G\), or equivalently \(H\), has the same property. Moreover \(D\) is said semisimple if \(H\) is semisimple.
Let \(Q\) be a quiver with vertices \(Q_0\) and arrows \(Q_1\). A representation of \(Q\) is a set of vector space \(\{V_i\}_{i\in Q_0}\) plus a set of morphisms associated to the arrows of \(Q\). Let \(V=\bigoplus V_i\), \(d=(\dim\,V_i)\) and \(\mathrm{Rep}(Q,d)=\bigoplus_{i\rightarrow j\in Q_1}\Hom(\mathbb C^{d_i},\mathbb C^{d_i})\). Moreover, let \(\mathrm{PGL}(Q,d)\) be the quotient of \(\mathrm{GL}(Q,d):=\bigoplus_{d_i\in Q_0}\mathrm{GL}_{d_i}\) by \(\mathbb C^*\mathrm{id}\). One can identify the \(Q\)-representations with dimension vector \(d\) with the points of \(\mathrm{Rep}(Q,d)\). Furthermore, one can define an action of \(\mathrm{PGL}(Q,d)\) over \(\mathrm{Rep}(Q,D)\), such that two representations are isomorphic if and only if they are in the same \(\mathrm{PGL}(Q,d)\) orbit. One say that \((Q,d)\) defines a linear free divisor \(D\), and call \(D\) a quiver linear free divisor, if the discriminant of the action of \(\mathrm{GL}(Q,d)\) on \(\mathrm{Rep}(Q,d)\) is a linear free divisor \(D\) with associated group \(G:=\mathrm{PGL}(Q,d)\). Such divisors are all reductive.
The main results about the first cited question can be summarized as follows (see Theorem 1.1, Theorem 2.8 and Theorem 2.9): 1) For a reductive linear free divisor, the number of its irreducible components coincides with the number of irreducible summands of the corresponding prehomogeneous space (and with the dimension of the center of \(G\)). 2) The semisimple linear free divisors can be explicitly classified up to castling transformations. 3) The abelian linear free divisors are the normal crossing divisors. 4) Quiver linear free divisors occur only in representation spaces of quivers without cycles.
In [Nagoya Math. J. 65, 1--155 (1977; Zbl 0321.14030)], \textit{M. Sato} and \textit{T. Kimura} introduce the notion of castling transformation and classify all the the irreducible prehomogeneous vector spaces up to castling transformations. Roughly speaking, two representations \((G_1,V_1)\) and \((G_2,V_2)\) are said to be castling transformations of one another if they are obtained by a third representation \((G_3,V_3)\) by tensoring \((G_3,V_3)\) (resp. \((G_3,V_3^*)\)) with the standard representation of two special linear groups \(\mathrm{SL}\). The authors use this result to give the cited explicit classification of irreducible reductive (that is, semisimple) linear free divisors.
The authors give also a characterization of quiver linear free divisors in terms of \(\mathrm{End}_Q(M)\) and \(\mathrm{Ext}_Q(M)\) (see Theorem 3.4).
\
The second question is motivated by the following result and conjecture. Denote by \(j:U=V\setminus D\rightarrow V\) the inclusion. One says that the logarithmic comparison theorem holds for \(D\) if the de Rham morphism \(\Omega_V^\bullet(\log\,D)\rightarrow j_*\Omega_V^\bullet\cong \mathbf{R}j_*\mathbb C_U\) is a quasi-isomorphism. The name is motivated by Grothendieck's comparison theorem which asserts that the de Rham morphism \(\Omega_V^\bullet(*D)\rightarrow j_*\Omega_V^\bullet\cong \mathbf{R}j_*\mathbb C_U\) is a quasi isomorphism.
A hypersurface is strongly Euler homogeneous if, for each \(p\in D\), \(G_p^o\nsubseteq G_f^o\), where \(f\) is any local defining equation of \(D\) at \(p\). A linear free divisor \(D\) in \(V\) is locally weakly quasihomogeneous if, for each \(p\in D\), the normal representation \(G_p\rightarrow \mathrm{GL}(T_pV/T_p(Gp))\) is non-negative, namely the convex hull of its weights does not contain 0.
In [Contemporary Mathematics 474, 245--269 (2008; Zbl 1166.32006)], \textit{L. Narváez-Macarro} proved that a locally weakly quasihomogeneous free divisor satisfies the logarithmic comparison theorem. Moreover it is conjectured that if a free divisor satisfies the logarithmic comparison theorem, then it is strongly Euler homogeneous (see also \textit{M. Granger, M. Schulze}, [Compos. Math. 142, No. 3, 765--778 (2006; Zbl 1096.32016)]).
The authors prove that all quiver linear free divisors are strongly Euler homogeneous (see Theorem 1.3). Moreover all quiver linear free divisors associated to a tame quiver are locally weakly quasihomogeneous, so the logarithmic comparison theorem holds for such divisors (see Corollary 1.4). A quiver is said tame if the Tits form \(q_Q\) is positive and non-definite. The Tits form is a quadratic form on \(\mathbf{Q}^{Q_0}\), which is in particular equal to \(q_Q(m)=\dim \mathrm{End}_Q(M)-\dim\;\mathrm{Ext}_Q(M)\) whenever \(m\) is the dimension vector of a \(Q\)-representation \(M\).
Finally, the authors consider reflections functors. These functors, introduced by \textit{I. N. Bernstein, I. M. Gel'fand} and \textit{V. A. Ponomarev} in [Russ. Math. Surv. 28, No.2, 17--32 (1973; Zbl 0279.08001), Reprint of Usp. Mat. Nauk 28, No. 2(170), 19--33 (1973; Zbl 0269.08001)], and independently as castling transforms by Sato and Kimura, play an important role in the study of quiver representations. The authors prove, in particular, that the reflection functors preserve the class of quiver linear free divisors (see Theorem 1.5). linear free divisor; prehomogeneous vector space; quiver representation; de Rham cohomology Granger, M; Mond, D; Schulze, M, Free divisors in prehomogeneous vector spaces, Proc. Lond. Math. Soc. (3), 102, 923-950, (2011) Homogeneous spaces and generalizations, Representation theory for linear algebraic groups, Lie algebras of vector fields and related (super) algebras, de Rham cohomology and algebraic geometry Free divisors in prehomogeneous vector spaces
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(F\) be the algebraic closure of the finite field with \(q\) elements and let \(\mathcal Q\) be a quiver with the underlying graph of Dynkin type \(A_n\). The group \(G_d=\text{GL}_{d_1}(F)\times\cdots\times \text{GL}_{d_n}\) acts by conjugation on \(E_d=\bigoplus_{i\to j\in{\mathcal Q}}\Hom_F(F^{d_i},F^{d_j})\). In the paper under review the authors describe the \(G_d\)-orbits \(\mathcal O\) with the property that the orbit closure \(\overline{\mathcal O}\) (in the Zariski topology) is rationally smooth. The approach is to consider the corresponding quantized enveloping algebra and to study the action of the bar involution on PBW bases. Then the authors use Ringel's Hall algebra approach to quantized enveloping algebras and Auslander-Reiten quivers, and describe the commutation relations between root vectors. As a result they obtain explicit formulas for the multiplication of an element of PBW bases adapted to a quiver with a root vector as well as recursive formulas to study the bar involution on PBW bases. As a consequence the authors derive that if the orbit closure is rationally smooth, then it is smooth.
The recent paper [\textit{P. Caldero} and \textit{R. Schiffler}, Ann. Inst. Fourier 54, No. 2, 265--275 (2004; Zbl 1126.17013)] contains the characterization of the rationally smooth orbit closures of representations of quivers of type \(A\), \(D\) or \(E\). Comparing the methods of both papers, the present paper has the advantage that the approach is very explicit and recursive, and may be used for computer programming. But it cannot easily be generalized to type \(D\) and \(E\). representations of quivers; varieties of representations; rational smoothness; quantum groups; quantized enveloping algebra Robert Bédard and Ralf Schiffler, Rational smoothness of varieties of representations for quivers of type \({A}\), preprint. Quantum groups (quantized enveloping algebras) and related deformations, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities in algebraic geometry, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Rational smoothness of varieties of representations for quivers of type \(A\)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\) be a smooth projective variety of dimension \(n\) and let \(D\) and \(D'\) be pseudoeffective divisors on \(X\). Recall that two divisors \(D\) and \(D'\) are numerically equivalent (\(D\equiv D'\)) if and only if \(D\cdot C=D'\cdot C\) for every irreducible curve \(C\) on \(X\). Let \(D=P+N\) be the divisorial Zariski decomposition. For a fixed point \(x\in X\), we can further decompose the negative part \(N=N_x+N_x^c\) into the effective divisors \(N_x\) and \(N_x^c\) such that every irreducible component of \(N_x\) passes through \(x\). We say that the decomposition
\[
D=P+N_x+N_x^c
\]
is the refined divisorial Zariski decomposition of \(D\) at a point \(x\).
We say that two pseudoeffective divisors \(D\) and \(D'\) are numerically equivalent near \(x\) (\(D\equiv_x D'\)) if \(P\equiv P'\) and \(N_x\equiv N_x'\). Now we decompose the divisor \(N_x\) as
\[
N_x=N_x^{\mathrm{sm}}+N_x^{\mathrm{sing}}
\]
where every irreducible component of \(N_x^{\mathrm{sm}}\) (resp. \(N_x^{\mathrm{sing}}\)) is smooth (resp. singular).
Let \(f: \widetilde{X}\rightarrow X\) be a birational morphism between smooth projective varieties of dimension \(n\) and let \(x\in X\) be a point. An admissible flag \(\widetilde{Y}_{\bullet}\) on \(\widetilde{X}\) is said to be
\begin{itemize}
\item centered at \(x\) if \(f(\widetilde{Y_n})=\{x\}\),
\item proper over \(X\) if \(\mathrm{codim} \, f(\widetilde{Y_i})=i\),
\item infinitesimal over \(X\) if \(\mathrm{codim} \, f(\widetilde{Y_n})=1\).
\end{itemize}
An admissible flag \(\widetilde{Y}_{\bullet}\) on \(\widetilde{X}\) that is proper over \(X\) is said to be induced (by an admissible flag \(Y_{\bullet}\) on \(X\)) if \(f(\widetilde{Y}_i)=Y_i\) for each \(0\leq i\leq n\).
An admissible flag \(\widetilde{Y}_{\bullet}\) on \(\widetilde{X}\) that is infinitesimal over \(X\) is said to be induced (by an admissible flag \(Y_{\bullet}\) on \(X\)) if there is a proper admissible flag \(Y_{\bullet}'\) on \(\widetilde{X}\) induced by \(Y_{\bullet}\) such that \(f(\widetilde{Y}_1)=Y_n\) and \(\widetilde{Y}_i=\widetilde{Y}_1\cap Y_{i-1}'\) for \(2\leq i\leq n\).
Let \(\Delta_{Y_{\bullet}}(D)\) be the Okounkov body of a divisor \(D\) with respect to an admissible flag \(Y_{\bullet}\).
The two main theorems of the paper give a generalization of Roe's results into higher dimensions.
{Theorem A.} Let \(D\) and \(D'\) be big divisors on a smooth projective varieties \(X\), and let \(x\in X\) be a point. Then the following are equivalent:
\begin{enumerate}
\item \(D\equiv_x D'\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for every admissible flag \(\widetilde{Y}_{\bullet}\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for every proper admissible flag \(\widetilde{Y}_{\bullet}\) over \(X\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for every infinitesimal flag \(\widetilde{Y}_{\bullet}\) over \(X\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\).
\end{enumerate}
{Theorem B.} Let \(D\) and \(D'\) be a big divisors on a smooth projective variety \(X\). For a fixed point \(x\in X\), consider the decompositions \[D=P+N_x^{\mathrm{sm}}+N_X^{\mathrm{sing}}+N_x^c, \quad D'=P'+{N'}_x^{\mathrm{sm}}+{N'}_x^{\mathrm{sing}}+{N'}_x^c.\] Then the following are equivalent
\begin{enumerate}
\item \(P\equiv P'\), \(N_x^{\mathrm{sm}}={N'}_x^{\mathrm{sm}}\), and \(\Delta_{Y_{\bullet}}(N_x^{\mathrm{sing}})=\Delta_{Y_{\bullet}}({N'}_x^{\mathrm{sing}})\) for every admissible flag \(Y_{\bullet}\) centered at \(x\),
\item \(\Delta_{Y_{\bullet}}(D)=\Delta_{Y_{\bullet}}(D')\) for every admissible flag \(Y_{\bullet}\) on \(X\) centered at \(x\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for every induced proper admissible flag \(\widetilde{Y}_{\bullet}\) on \(X\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for almost every induced infinitesimal admissible flag \(\widetilde{Y}_{\bullet}\) on \(X\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\).
\end{enumerate} Okounkov bodies; numerical equivalences; Zariski decomposition Divisors, linear systems, invertible sheaves, Convex sets in \(n\) dimensions (including convex hypersurfaces) Local numerical equivalences and Okounkov bodies in higher dimensions
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(S\subseteq \mathbb{N}\) be a semigroup generated by a generalized arithmetic sequence, that is \(S=\langle m_0, hm_0+d,hm_0 + 2d, \dots , hm_0+nd \rangle \), where \((m_0,d)=1\) and \(h\geq 1\). To \(S\) one can associate an affine monomial curve over a field \(k\) with coordinate ring \(R=k[t^s\mid s\in S]\). In this paper the authors compute the minimal free resolution of \(R\), extending a previous result in [\textit{P. Gimenez} et al., J. Algebra 388, 294--310 (2013; Zbl 1291.13021)] for arithmetic sequences (\(h=1\)). Moreover, for a large class of semigroups generated by arithmetic sequences, they give smooth deformations of the matrices in the minimal free resolution, which they use to prove the Weierstrass property for these semigroups. numerical semigroup; generalized arithmetic sequence; monomial curve; free resolution; deformation; Weierstrass semigroup Syzygies, resolutions, complexes and commutative rings, Commutative semigroups, Riemann surfaces; Weierstrass points; gap sequences, Semigroup rings, multiplicative semigroups of rings Syzygies of \textit{GS} monomial curves and Weierstrass property
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A general ROTH system \(S=(\ell_1,\dots,\ell_n,c(1),\dots,c(n))\) is given by \(n\) linearly independent linear forms \((\ell_1,\dots,\ell_n)\) in \(n\) variables with real algebraic coefficients and \(n\) real numbers \(c(1),\dots,c(n)\) such that \(c(1)+\dots+c(n)=0\) and such that, for each \(\delta>0\), there are only finitely many integral solutions to the system of inequalities
\[
|\ell_q|<Q^{-c(q)-\delta}, \quad Q>1, \; 1\leq q\leq n.
\]
The special case \(n=2\), \(\ell_1(x_1,x_2)=x_1\), \(\ell_2(x_1,x_2)=\alpha x_1-x_2\) where \(\alpha\) is an irrational real algebraic number and \(c(1)=-1\), \(c(2)=1\) is called a classical ROTH system.
To a general ROTH system \(S\) is attached a filtration \(F_S^\cdot V\) over \(\overline{\mathbb Q}\) of the \(\mathbb Q\)-vector space \(V=\mathbb Q x_1+\dots + \mathbb Q x_n\), defined as
\[
F_S^{i}V=\sum_{c(q)\geq i}\overline{\mathbb Q} \ell_q \quad (i\in\mathbb R).
\]
In the case of a classical ROTH system attached to an irrational algebraic number \(\alpha\), this filtered space is denoted by \((\check{V}, F_\alpha^\cdot \check{V})\). The slope of a filtered vector space \((V, F^\cdot V)\) is
\[
\mu(V)=\frac{1}{\dim_\mathbb Q V} \sum_{w\in \mathbb R} w \dim_{\overline{\mathbb Q}} {\mathrm{gr}}^w(F^{\cdot} V)
\]
where \( {\mathrm{gr}}^w(F^{\cdot} V)=F^wV/F^{w+}V\), \(F^{w+}V=\bigcup _{j>w}F^jV\). The filtration is semi stable if \(\mu(W)=\mu(V)\) for any nonzero \(\mathbb Q\)-vector subspace \(W\) of \(V\). The category of finite dimensional vector spaces over \(\mathbb Q\) with semistable filtration of slope zero is denoted by \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\). See Chap.~VI, Theorem 2B of [\textit{W. M. Schmidt}, Diophantine approximation. Berlin etc.: Springer-Verlag (1980; Zbl 0421.10019)]. See also [\textit{G. Faltings}, Proc. ICM '94. Vol. I. Basel: Birkhäuser, 648--655 (1995; Zbl 0871.14010)] and [\textit{B. Totaro}, Duke Math. J. 83, No. 1, 79--104 (1996; Zbl 0873.14019)].
Here is the main result of the paper under review. Let \(\alpha\) be an irrational real algebraic number. If \(\alpha\) is not quadratic, then there exists a fully faithful tensor functor \(\iota\) of the category \({\mathrm{Rep}}_\mathbb Q {\mathrm{SL}}_2\) of finite dimensional representations over \(\mathbb Q\) of the special linear group \({\mathrm{SL}}_2\) of degree \(2\) into the tensor category \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\) such that the functor \(\iota\) commutes with the forgetful tensor functor to the tensor category \({\mathrm{Vec}}_\mathbb Q\) of finite dimensional vector spaces over \(\mathbb Q\) and such that the image of \(\iota\) contains the filtered vector space \((\check{V}, F_\alpha^\cdot \check{V})\) derived from a classical ROTH System.
On the other hand, if \(\alpha\) is quadratic, then there exists a fully faithful tensor functor \(\iota\) of the category \({\mathrm{Rep}}_\mathbb Q T_\alpha\) of finite dimensional representations over \(\mathbb Q\) of a one dimensional anisotropic torus \(T_\alpha\) over \(\mathbb Q\) into the tensor category \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\) such that the group \(T_\alpha(\mathbb Q)\) is isomorphic to the kernel of the norm map of the quadratic number field \(\mathbb Q(\alpha)\) over \(\mathbb Q\), such that the functor \(\iota\) is compatible with the forgetful tensor functor to \({\mathrm{Vec}}_\mathbb Q\) and such that the image of \(\iota\) contains the filtered vector space \((\check{V}, F_\alpha^\cdot \check{V})\). M. FUJIMORI, The algebraic groups leading to the Roth inequalities. J. TheÂor. Nombres Bordeaux, 24: pp. 257-292, 2012. Approximation to algebraic numbers, Simultaneous homogeneous approximation, linear forms, Rational points, Monoidal categories (= multiplicative categories) [See also 19D23] The algebraic groups leading to the Roth inequalities
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of \(m\)-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of \(r\) filtrations on an analytically unramified local ring \(R\) come from the coefficients of a suitable homogeneous polynomial in \(r\) variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of \(m\)-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for \(m\)-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals. mixed multiplicity; valuation; filtration; divisorial filtration Multiplicity theory and related topics, Valuations and their generalizations for commutative rings, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Multiplicities and mixed multiplicities of arbitrary filtrations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We closely follow the very clear and instructive introduction to the paper.
From the introduction:
``In commutative algebra and algebraic geometry, some of the most subtle problems arise in the context of mixed characteristic, i.e. over local fields such as \({\mathbb Q}_p\) which are of characteristic \(0\), but whose residue field \({\mathbb F}_p\) is of characteristic \(p\). The aim of this paper is to establish a general framework for reducing certain problems about mixed characteristic rings to problems about rings in characteristic \(p\). We will use this framework to establish a generalization of Faltings's almost purity theorem, and new results on Deligne's weight-monodromy conjecture.''
The point of departure is the following theorem of \textit{J.-M. Fontaine} and \textit{J.-P. Wintenberger} [C. R. Acad. Sci., Paris, Sér. A 288, 441--444 (1979; Zbl 0403.12018)]:
Theorem. The absolute Galois groups of \({\mathbb Q}_p (p^{1/p^{\infty}})\) and \({\mathbb F}_p((t))\) are canonically isomorphic.
The autzor defines a perfectoid field to be a complete topological field \(K\) whose topology is induced by a nondiscrete valuation of rank \(1\), such that the Frobenius is surjective on \(K^0/p\), where \(K^0\) denotes the subring of powerbounded elements in \(K\). Following the construction of Fontaine, to any such \(K\) one associates another perfectoid field \(K^{\flat}\) of characteristic \(p\). The above theorem is then generalized to:
Theorem. The absolute Galois groups of \(K\) and \(K^{\flat}\) are canonically isomorphic.
The objective is then to generalize this to a comparison of geometric objects over \(K\) with geometric objects over \(K^{\flat}\). In order to adequately formulate such a generalization in turns out that Huber's \textit{adic spaces} are a suitable framework. For example, any variety over \(K\) has an associated adic space \(X^{\text{ad}}\) over \(K\) with underlying topological space \(|X^{\text{m ad}}|\).
Theorem. There is a homeomorphism of topological spaces
\[
|({\mathbb{A}}^1_{K^{\flat}})^{\text{ad}}|\cong\text{lim}_{\leftarrow\atop T\mapsto T^p} |({\mathbb{A}}^1_{K})^{\text{ad}}|
\]
(where \(T\) is a coordinate on the affine line).
Next it is argued that also the respective structure sheaves can be compared --- although their characteristics (i.e., \(\text{Char}(K)\in\{0,p\}\) versus \(\text{Char}(K^{\flat})=p\)) may differ! The author defines a perfectoid \(K\)-algebra (over the perfectoid field \(K\)) to be a Banach \(K\)-algebra \(R\) such that the set of power bounded elements \(R^0\subset R\) is bounded, and such that the Frobenius is surjective on \(R^0/p\). He proves:
Theorem. There is a natural equivalence of categories, called the tilting equivalence, between the category of perfectoid \(K\)-algebras and the category of perfectoid \(K^{\flat}\)-algebras.
An example of a perfectoid algebra is \(K\langle T^{\frac{1}{p^{\infty}}}\rangle\), its tilt is \(K^{\flat}\langle T^{\frac{1}{p^{\infty}}}\rangle\).
Now affinoid perfectoid spaces are defined --- they are associated with pairs \((R,R^+)\), where \(R\) is a perfectoid \(K\)-algebra and \(R^+\subset R^0\) is open and integrally closed (often \(R^+=R^0\)). Proceeding as in Huber's definition of adic spaces, the author now defines perfectoid spaces, built from affinoid perfectoid spaces.
Theorem. The category of perfectoid spaces over \(K\) and the category of perfectoid spaces over \(K^{\flat}\) are equivalent.
In order to define and investigate the étale site \(X_{\text{ét}}\) of a perfectoid space \(X\), a generalization of Faltings's almost purity theorem is needed:
Theorem. Let \(R\) be a perfectoid \(K\)-algebra. Let \(S/R\) be finite étale. Then \(S\) is a perfectoid algebra, and \(S^0\) is almost finite étale over \(R^0\).
This theorem essentially compares finite étale covers of a perfectoid space \(X\) with those of \(X^{\flat}\) and leads to
Theorem. Let \(X\) be a perfectoid space over \(K\), with tilt \(X^{\flat}\) over \(K^{\flat}\). Then tilting induces an equivalence of sites \(X_{\text{ét}}\cong X^{\flat}_{\text{ét}}\).
Theorem. The étale topos \(({\mathbb{P}}^{n,\text{ad}}_{K^{\flat}})^{\widetilde{\,}}_{\text{ét}}\) is equivalent to the inverse limit \(\text{lim}_{\leftarrow \varphi}({\mathbb{P}}^{n,\text{ad}}_{K})^{\widetilde{\,}}_{\text{ét}}\).
Finally, applications of these results to the weight monodromy conjecture are given.
Theorem. Let \(k\) be a local field of characteristic \(0\). Let \(X\) be a geometrically connected proper smooth variety over \(k\) such that \(X\) is a set theoretic complete intersection in a projective smooth toric variety. Then the weight monodromy conjecture is true for \(X\). perfectoid space; perfectoid algebra; perfectoid field; almost mathematics; adic spaces; purity theorem; weight-monodromy conjecture Scholze, Peter, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci., 116, 245-313, (2012) Étale and other Grothendieck topologies and (co)homologies, Galois theory, Local ground fields in algebraic geometry, Rigid analytic geometry, Varieties over finite and local fields Perfectoid spaces
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(L\) be an ample line bundle on a nonsingular projective variety \(X\) and \(V \subseteq H^0(L)\) a subvector space defining an embedding \(\varphi_V: X\hookrightarrow \mathbb{P}(V^*) = \mathbb{P}_N\). Then \(S=\bigoplus_{k\geq 0} S^kV\) is the homogeneous coordinate ring of \(\mathbb{P}_N\). Define the graded \(S\)-module \(R=R(V) = \bigoplus_{k\geq 0} R_k\), with \(R_k=K\) if \(k=0\), \(R^k=V\) if \(k=1\), \(R_k=H^0(L^k)\) if \(k\geq 2\), and let
\[
0\to \bigoplus_j S(-j)^{\widetilde b_{rj}} \to \cdots \to\bigoplus_j S(-j)^{\widetilde b_{0j}} \to R\to 0
\]
be a minimal free resolution of \(R\) with graded Betti numbers \(\widetilde b_{ij}\).
Definition. For an integer \(p\geq 0\) the vector space \(V\) (respectively its associated embedding \(\varphi_V: X\hookrightarrow \mathbb{P}_N)\) is said to satisfy the property \(\widetilde N_p\), if \(\widetilde b_{0,j}=1\) if \(j=0\), \(\widetilde b_{0,j}=0\) otherwise, and \(\widetilde b_{ij}=0\) for \(1\leq i\leq p\), and \(j>i+1\). For a complete linear system, i.e. \(V=H^0(L)\), the property \(\widetilde N_p\) coincides with the property \(N_p\) introduced by \textit{M. Green} and \textit{R. Lazarsfield} [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008) and Compos. Math. 67, No. 3, 301-314 (1988; Zbl 0671.14010)].
In this paper the properties \(\widetilde N_p\) are investigated for noncomplete Veronese embeddings. It is shown that the linear projection of a complete Veronese embedding \(\mathbb{P}_n \hookrightarrow \mathbb{P}_N\) from a general linear subspace of \(\mathbb{P}_N\) of low dimension satisfies property \(\widetilde N_0\). Moreover, theorem 3.1 yields an upper bound for \(p\) for a noncomplete Veronese embedding of the projective plane \(\mathbb{P}_2\) to satisfy property \(\widetilde N_p\). linear systems; ample line bundle; graded Betti numbers; noncomplete Veronese embeddings Christina Birkenhake, Linear systems on projective spaces, Manuscripta Math. 88 (1995), no. 2, 177 -- 184. Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves, Embeddings in algebraic geometry Linear systems on projective spaces
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R, \mathfrak{m})\) be a regular local ring and \(y, u_1, \ldots, u_d\) be a regular system of parameters. Let \(f\in \mathfrak{m}\) and \(f\not\in \langle u_1, \ldots, u_d\rangle\). Hironaka associated a polyhedron \(\Delta(f; u_1, \ldots, u_d;y)\subset \mathbb{R}^d_{\geq 0}\) and proved that there exists \(z\in\widehat{R}\) such that \(\{z, u_1, \ldots, u_d\}\) is a regular system of parameters of \(\widehat{R}\) and
\[
\Delta(f; u_1, \ldots, u_d; z)=\underset{(\widehat{y}, u_1, \ldots, u_d)}{\bigcap} \Delta(f; u_1, \ldots, u_d, \widehat{y})
\]
where the intersection is over all regular systems of parameters of \(\widehat{R}\) of the form \((\widehat{y}, u_1, \ldots, u_d)\), (c.f. \textit{H. Hironaka} [J. Math. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)]).
It is proved that \(z\) can be chosen in \(R\) in case \(R\) is a \(G\)--ring (i.e. the formal fibres are geometrically regular). characteristic polyhedron of a singularity; Hironaka Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Characteristic polyhedra of singularities without completion
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\) be a non-archimedean real closed field. Denote its valuation ring by \(V\) and its residue field by \(\overline R\). The paper studies reduction of definable subsets \(S\) of \(V^n\), i.e. subsets of \(V^n\) that are defined by a formula in the first-order language of ordered valuation fields. By the reduction of \(S\) is meant the image \(\overline S\) of \(S\) under the residue map \(V^n \to \overline R^n\).
The author gives a geometric proof of the (known) fact that \(\overline S\) is semi-algebraic whenever \(S\) is definable. He defines a reduction map \(r : H_p (S,T) \to H_p (\overline S, \overline T)\) on relative homology groups with coefficients in \(\mathbb{Z}/2\mathbb{Z}\) for a pair of semi- algebraic subsets \(T \subseteq S\) of \(V^n\). Functoriality of this reduction map with respect to so-called moderate maps is proved. Finally, the author proves that this reduction map behaves well with respect to the mod-2 intersection product. semi-algebraicity of definable subsets; non-archimedean real closed field; valuation fields L. Bröcker, On the reduction of semialgebraic sets by real valuations, in: ``Recent advances in real algebraic geometry and quadratic forms'', Contemp. Math., 155, Amer. Math. Soc., Providence, RI, 1994, pp. 75-95. Zbl0826.14038 MR1260702 Semialgebraic sets and related spaces, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Non-Archimedean valued fields, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Valuation rings On the reduction of semialgebraic sets by real valuations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The starting point of this paper is the classification of \(p\)-divisible groups over a discrete valuation ring of characteristic 0 with perfect residue field of characteristic \(p\geq3\) obtained by \textit{C.~Breuil} [C.~R.~Acad.~Sci. Paris Sér.~I Math. 328, 93--97 (1999; Zbl 0924.14025)]. Using the author's new Dieudonné theory [Astérisque No. 278, 127--248 (2002; Zbl 1008.14008)], the author shows how the similar classification for formal \(p\)-divisible groups can be extended over a much more general base ring. The classification for arbitrary \(p\)-divisible groups is also given but over a less general base ring.
We now describe the main results of this paper. Let \(p\) be a fixed rational prime. Let \(R\) be a commutative ring in which \(p\) is nilpotent. A frame for \(R\) is a triple \((A,J,\sigma)\), where
(a) \(A\) is a \(p\)-adic ring (separated and complete for the \(p\)-adic topology) which is torsion-free as an abelian group, (b) \(\sigma\) is an endomorphism on \(A\) that induces the Frobenius map on \(A/pA\),
(c) there is a surjective ring homomorphism \(A\to R\) whose kernel \(J\) has divided powers.
Definition. (1) A Dieudonné A-window over \(R\) relative to a frame \((A,J,\sigma)\) is a triple \((M,M_1,\Phi)\), where (a) \(M\) is a finitely generated projective \(R\)-module, (b) \(M_1\subset M\) is an \(A\)-submodule containing \(JM\), (c) \(\Phi\) is a \(\sigma\)-linear map on \(M\), such that (i) \(M/M_1\) is a projective \(R\)-module, and (ii) \(\Phi M_1\subset pM\) (so \(\frac{1}{p}\Phi M_1\subset M\)) and \(M\) is generated by \(\Phi M\) and \(\frac{1}{p}\Phi M_1\) over \(A\).
(2) Let \((M,M_1,\Phi)\) be a Dieudonné \(A\)-window over \(R\). Then there is a unique \(A\)-linear map \(\Psi:M\to A\otimes_{\sigma,A}M\) such that \(\Psi(\Phi(m))=p\otimes m\) for all \(m\in M\). \((M,M_1,\Phi)\) is called an \(A\)-window if \(\Psi^N(M)\subset J\otimes_{\sigma^N,A}M\) for some positive integer \(N\).
Theorem. (1) Let \(R\) be an excellent ring. Then the category of \(A\)-windows over \(R\) is equivalent to the category of formal \(p\)-divisible groups over \(R\).
(2) Let \(R\) be an artinian ring with perfect residue field of characteristic \(p\geq 3\). Then the category of Dieudonné \(A\)-windows over \(R\) (relative to a more restricted frame) is equivalent to the category of \(p\)-divisible groups over \(R\). Dieudonné theory; displays; Dieudonné A-windows Zink, Th., Windows for displays of p-divisible groups, 491-518, (2001), Birkhäuser, Basel Formal groups, \(p\)-divisible groups, Finite ground fields in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology Windows for displays of \(p\)-divisible groups
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We prove the following Lefschetz-type result:
Let \((A, {\mathfrak M})\) be an excellent, local, normal domain, \(t \in {\mathfrak M}\) such that \(A/tA\) is a domain satisfying the regularity condition \(R_1\), and \(B\) be the integral closure of \(A/tA\). If \(e \in \mathbb{Z}\), \(e > 1\), is a unit in \(A\), then the kernel, \(\ker \theta\), of the homomorphism \(\theta : \text{Cl} (A) \to \text{Cl} (B)\) contains no element of order \(e\).
An example of the above map \(\theta\) is given with \(\ker\theta\) finitely generated, torsion-free. -- Suppose \((R, {\mathfrak N}, k)\) to be an excellent, local, normal domain. Letting \(A : = R[[T]]\) and \(t = T\), in the above setting, allows us to establish conditions on \(R\) such that \(R\) has discrete divisor class group (DCG), i.e., \(\ker \theta = 0\). The ascent and descent of the DCG property is investigated for generically Galois extensions \(B \subset A\).
Finally, we show that for a local, normal domain, \((A, {\mathcal M})\), which is approximation and a \(\mathbb{Q}\)-algebra, there is an isomorphism, \(\text{Cl} (A)_t \cong \text{Cl} (\widehat A)_t\), between the torsion subgroups of the divisor class groups of \(A\) and its \({\mathfrak M}\)-adic completion \(\widehat A\). \(R_ 1\); DCG; integral closure; discrete divisor class group; ascent; descent; Galois extensions; normal domain Griffith P., J. Algebra 167 (2) pp 473-- (1994) Class groups, Picard groups, Hypersurfaces and algebraic geometry, Galois theory and commutative ring extensions Restrictions of torsion divisor classes to hypersurfaces
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The aim of this paper is to give some relationships between the Riemann-Zariski space of the local ring of an algebraic variety \(X\) at a point \(x\) and the geometry of \(X\) at \(x\). The Riemann-Zariski space of \(X\) at \(x\) is the set of valuations of the fraction field of \(X\) dominating \(\mathcal{O}_{X,x}\) endowed with the Zariski topology.
The first result of this note is the fact that, for two regular points \(x\) and \(y\) of two algebraic varieties \(X\) and \(Y\) defined over the same algebraically closed field, the Riemann-Zariski spaces of \(X\) at \(x\) and \(Y\) at \(y\) are homeomorphic if and only if the dimensions of \(X\) and \(Y\) are the same.
The second result concerns the normal surfaces case: for two points \(x\) and \(y\) of two normal surfaces \(X\) and \(Y\) defined over the same algebraically closef field, the Riemann-Zariski spaces of \(X\) at \(x\) and \(Y\) at \(y\) are homeomorphic if and only if the resolution graphs of the germs \((X,x)\) and \((Y,y)\) are equivalent. This notion of equivalence of graphs involves the notion of the core of the graphs and this equivalence is stronger than the homotopical equivalence.
For both statements it is also shown that the Riemann-Zariski space of \((X,x)\) can be replaced by the normalized non-Archimedean link of \((X,x)\) defined in the theory of Berkovich analytic spaces. valuations; Riemann-Zariski space; resolution of singularities; normal surfaces Valuations and their generalizations for commutative rings, Rigid analytic geometry, Singularities of surfaces or higher-dimensional varieties On the homeomorphism type of some spaces of valuations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system For a finite set \(H \subset {\mathbb Z}^n\), let \({\mathbb C} [H]\) be the space of all Laurent polynomials with complex coefficients whose support are contained in \(H\). For a collection \(A = (A_0, \ldots, A_k)\) of finite subsets of \(\mathbb Z^n\), \({\mathbb C}[A]\) denotes the space \({\mathbb C}[A_0] \oplus \cdots \oplus {\mathbb C}[A_k].\) Each system of \(k+1\) polynomial equations in \(n\) variables can be considered as an element of \(C[A]\). The author considers various questions related to such system of polynomial equations, and seek answers through the combinatorics of the tuple \(A\) and the topology of \(f\).
As a start, he defines and characterizes a degenerate system in Theorem 1.1. Under the relevant condition for the tuple \(A\) (Definition 1.3), Theorem 1.4 shows that the set of all degenerate systems is a non-empty hypersurface. The author then consideres the notion of Euler discriminant and Bertini discriminant and gave a formula for them. As an application, he characterizes bifurcation sets, and applies this to the study of topology of polynomial mappings. The results of this paper generalize classical results about resultants, \(A\)-discriminants, and sparse discriminants. It also opens up a bunch of interesting conjectures relating to the notion of discriminants. discriminant; bifurcation set; topology of polynomial mapping Esterov, A., The discriminant of a system of equations, Adv. Math., 245, 534-572, (2013) Solving polynomial systems; resultants, The discriminant of a system of equations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(A\) be a 3-dimensional excellent henselian normal local ring. Assume that the residue field \(K\) of \(A\) is algebraically closed or finite. Suppose also that the unique singular point of \(A\) is the closed point \(x\). Let \(f: X \rightarrow \text{Spec } A \) be a resolution of \(\text{Spec } A\) where \(X\) is a 3-dimensional regular scheme. Denote by \(Y\) the surface \(f^{-1} (x)\) and by \(P\) the set of height 1 prime ideals of \(A\). Denote \(Q_{l} (i) = \bigotimes_{j=1}^{i} Q_{j}\), \(Q_{j}\) being the Tate module \(Q_{l}\) for each \(j\).
The paper is basically devoted to prove two results. Firstly, when \(K\) is algebraically closed and of characteristic 0, the mapping \(H^{3} (K, Q_{l} (2)) \rightarrow \bigoplus_{v \in P} H^{3} (K_{v}, Q_{l} (2))\) contains a subgroup isomorphic to \(Q_{l}^{r_{1} (A)}\), where \(l\) is a prime number, \(K_{v}\) the completion of \(K\) in \(v\) and \(r_{1} (A)\) the dimension of the set of elements in \(H^{2} (Y, Q_{l})\) with weight \(\leq 1\). Secondly, if \(K\) is a finite field of characteristic \(p \neq 2, 3, 5\), the authors consider the mapping \(H^{4} (K, Q_{l} (3)) \rightarrow \bigoplus_{v \in P} H^{4} (K_{v}, Q_{l} (3))\) and prove that its kernel contains a subgroup of type \((Q_{l})^{r_{1}'} (A) \) where \(r_{1}' (A) = \text{rank}_{Q_{l}} H^{2} ( |\Gamma |, Q_{l})\), \(|\Gamma |\) being the geometric realization of the exceptional fiber \(Y\). Moreover, the data \(r_{1}' (A) \) and \(r_{1} (A)\) are independent of the resolution \(f\) of \(X\).
Similar results when \(\dim A = 2\) are also given generalizing a previous result by \textit{S. Saito} [in: Galois representations and arithmetic algebraic geometry, Proc. Symp., Kyoto 1985 and Tokyo 1986, Adv. Stud. Pure Math. 12, 343-373 (1987; Zbl 0672.12006)]. excellent henselian normal local ring; Tate modules; exceptional fibre Draouil B and Douai J C, Sur l'arithmétique des anneaux locaux de dimension 2 et 3, J. Algebra 213 (1999) 499--512 Henselian rings, Galois cohomology, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Étale and flat extensions; Henselization; Artin approximation On the arithmetic of local rings of dimension 2 and 3
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Suppose that \(S=\bigoplus_{d\geqslant 0}S_d\) is a graded associative algebra generated by \(S_1\) with an automorphism \(\sigma\) of \(S_1\) as a linear space which can be extended to an automorphism \(\sigma\) of \(S\). Then there is a twisted algebra \(S^\sigma\) with multiplication defined as \(a*b=a\cdot\sigma(b)\) for homogeneous elements \(a,b\in S\).
The author considers the case when \(S\) is the complex polynomial algebra on \(Y_1,\dots,Y_n\) and \(\sigma\) is defined as \(\sigma(Y_1)=Y_1\) and \(\sigma(Y_{i+1})=Y_{i+1}+Y_i\) for \(i>0\). There is given a classification of the primitive ideals in \(S^\sigma\) which do not contain \(Y_1\). In particular, every primitive ideal in \(S^\sigma\) is generated by a regular sequence of homogeneous elements \(g_1,\dots,g_t\) where each \(g_i\) is irreducible and \(\sigma\)-invariant modulo the ideal generated by the preceding elements. So there exists a one to one correspondence between primitive ideals of \(S^\sigma\) and symplectic leaves of the Poisson structure induced by \(\sigma\). This result can be generalized to the case when an automorphism of \(S\) has a single eigenvalue. In this case the leaves are algebraic and realizable by orbits of an algebraic group. deformations of algebras; primitive ideals; twisted homogeneous coordinate rings; symplectic leaves; Poisson manifolds; complex polynomial rings; Poisson structures; primitive spectra Deformations of associative rings, Ordinary and skew polynomial rings and semigroup rings, Simple and semisimple modules, primitive rings and ideals in associative algebras, Graded rings and modules (associative rings and algebras), Ideals in associative algebras, Derivations, actions of Lie algebras, Noncommutative algebraic geometry Primitive and Poisson spectra of single-eigenvalue twists of polynomial algebras.
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For the entire collection see Zbl 0745.00042.]
The first section of this paper leads to an algebraic conjecture generalizing Green's conjecture.
Let \(S=k [x_ 0,\dots,x_ r]\), and let \(R=S/I\) be a homogeneous factor ring of \(S\). We assume that \(I\) contains no linear forms, and the projective dimension of \(S/I\) is \(m\). Then minimal free resolution \({\mathcal F}\) of \(S/I\) can be written as
\[
\begin{aligned} 0 \leftarrow S/I & \leftarrow S \leftarrow S(-2)^{a_ 1} \oplus S(-3)^{b_ 1} \oplus \cdots \leftarrow S(-3)^{a_ 2} \oplus S(-4)^{b_ 2} \oplus \cdots\\ & \leftarrow \cdots \leftarrow S(-(m+1))^{a_ m} \oplus S(- (m+2))^{b_ m} \oplus S(-(m+3))^{c_ m} \oplus \cdots \leftarrow 0,\end{aligned}
\]
with \(a_ i,b_ i \in \mathbb{Z}\), \(a_ i,b_ i \geq 0\). We define the 2-linear strand of \({\mathcal F}\) to be the subcomplex
\[
0 \leftarrow S/I \leftarrow S \leftarrow S(-2)^{a_ 1} \leftarrow S(- 3)^{a_ 2} \leftarrow \cdots \leftarrow S(-(m+1))^{a_ m} \leftarrow 0.
\]
The author defines the length of the 2-linear strand to be the largest number \(n\) such that \(a_ n\neq 0\). He calls this \(n\) the 2-linear projective dimension and writes \(2\text{LP}(S/I)=n\).
If \(I\) is the ideal generated by the \(2 \times 2\) minors of a generic \(p \times q\) matrix, then the 2-linear strand is known to have length \(\geq p+q-3\). As a form of converse the author is lead to the following algebraic conjecture: Let \(k\) be an algebraically closed field of characteristic \(\neq 2\), and let \(I \subset S=k [x_ 0,\dots,x_ r]\) be a prime ideal, containing no linear form, whose quadratic part is spanned by quadrics of rank \(\leq 4\). If \(2\text{LP}(S/I)=n\), then \(I\) contains an ideal of \(2 \times 2\) minors of a 1-generic \(p \times q\) matrix with \(p+q-3=n\).
Green's conjecture, from the algebraic point of view, is just the special case of this where (a) \(S/I\) is normal (= integrally closed); (b) \(\dim S/I=2\); (c) \(S/I\) is Gorenstein; (d) degree \(S/I = 2r\).
In section two the author considers the canonical ring of a non- hyperelliptic curve (= the homogeneous coordinate ring of the canonically embedded curve) and gets the geometric conjecture [\textit{M. L. Green}, Invent. Math. 75, 85-104 (1984; Zbl 0542.14018)]:
The length of the 2-linear part of the resolution \({\mathcal F}\) of the canonical ring \(S/I\) of a curve of genus \(g\) and Clifford index \(c\) is \(2\text{LP}(S/I)=g-2-c\). The generic form there of becomes:
The free resolution of the canonical ring of a generic curve of genus \(g\) has \(a_{\lfloor g/2 \rfloor},\dots,a_{g-3}=0\), \(0,\dots,0\).
The third section of the paper surveys some approaches to Green's conjecture. canonical ring of a non-hyperelliptic; minimal free resolution; 2-linear projective dimension; genus; Clifford index Eisenbud, D.: Green's conjecture: an orientation for algebraists, (Sundance, UT, 1990). Research Notes Mathematics, vol. 2, pp. 51-78. Jones and Bartlett, Boston, MA (1992) Curves in algebraic geometry, Complexes, Homological dimension and commutative rings, Special algebraic curves and curves of low genus Green's conjecture: An orientation for algebraists
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\tilde{\mathcal A}_ m\) be the algebra of functions on \(\mathbb{R}^ n\) generated by polynomial functions and exponentials of linear forms. The subset \(S\) in \(\mathbb{R}^ n\) belongs to \({\mathcal P}_ n\) if and only if there exist \(m\) and \(F\) in \(\tilde{\mathcal A}_{n+m}\) for which \(S\) is the image of the zerosubset of \(F\) by the canonical projection of \(\mathbb{R}^{n+m}\) onto \(\mathbb{R}^ n\). Let \(\tilde{\mathcal P}_ n\) be the smallest subset of parts in \(\mathbb{R}^ n\) which contains \({\mathcal P}_ n\), their closures and the images by the canonical projection of the elements in \(\tilde{\mathcal P}_{n+m}\). --- This family of sets is defined by an induction in two steps. The main goal of this article is to prove that \(\tilde{\mathcal P}_{n+m}\) contains the complementary part of each element in \(\tilde{\mathcal P}_{n+m}\), the union and the intersection of every finite family in \(\tilde{\mathcal P}_{n+m}\). The key results for the proof are theorems by A. G. Khovanskij on the set of the solutions of a Pfaff system on a Pfaff manifold. polynomial functions; exponential; zeroset of a function; projection; A. G. Khovanskij's theorems; Pfaff system; Pfaff manifold Charbonnel, J. -Y.: Sur certains sous-ensembles de l'espace euclidien. Ann. inst. Fourier Grenoble 41, No. 3, 679-717 (1991) Nash functions and manifolds, Linear function spaces and their duals, Pfaffian systems, Real-analytic and Nash manifolds, Relevant commutative algebra About certain subsets of the Euclidean space
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We show that a surface in \(\mathbb{P}^3\) parametrized over a 2-dimensional toric variety \(\mathcal S\) can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over \(\mathbb{P}^2\) established by \textit{L. Busé} and \textit{M. Chardin} [J. Symbolic Comput. 40, 1150--1168 (2005; Zbl 1120.14052)] and \textit{L. Busé} and \textit{J.P. Jouanolou} [J. Algebra 265, 312--357 (2003; Zbl 1050.13010)]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case \(\mathcal S=\mathbb{P}^1\times\mathbb{P}^1\) in detail and give numerous examples. matrix representation; rational surface; syzygy; approximation complex; implicitization; toric variety Nicolás Botbol, Alicia Dickenstein, and Marc Dohm, Matrix representations for toric parametrizations, Comput. Aided Geom. Design 26 (2009), no. 7, 757 -- 771. Computer-aided design (modeling of curves and surfaces), Syzygies, resolutions, complexes and commutative rings, Spline approximation, Computational aspects of algebraic surfaces, Toric varieties, Newton polyhedra, Okounkov bodies Matrix representations for toric parametrizations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(A\) be a commutative unitary ring, let \(F\) an \(A\)-module, and let
\[
0 \longrightarrow F @>\varepsilon>> K^0@>d^0>> K^1@>d^1>>\cdots
\]
be a resolution of \(F\). The resolution \(K^\cdot\) is said to be null-homotopic if there exists a family \(h^\cdot = (h^n)_n\) of morphisms \(h^n \colon K^n \longrightarrow K^{n-1}\) such that \(h^{n+1} d^n + d^{n-1} h^n = \text{id}_{K^n}\). If the homotopy \(h^\cdot\) also satisfies \(h^{n-1} h^n = 0\), then \(h^\cdot\) is said to be normal, and the pair \((K^\cdot, h^\cdot)\) is called a normal homotopic complex. The null-homotopic resolutions of \(F\) together with the sets of morphisms defined in the obvious way form a category, and if \(F\) admits a normal homotopic resolution \((L^\cdot, h^\cdot)\), then \(L^\cdot\) is an initial object in that category.
Let now \((X, {\mathcal O}_X)\) be a ringed space, and let \(\mathcal F\) be an \({\mathcal O}_X\)-module. A resolution \({\mathcal L}^\cdot\) of \(\mathcal F\) is said to be homotopic if it is null-homotopic stalk-wise. The homotopic resolutions of \(\mathcal F\) together with the obvious morphisms form again a category, and the canonical Godement resolution of \(\mathcal F\) is a final object in that category.
In the case of a smooth manifold \(X\), this approach allows the authors to construct explicitly a natural quasi-isomorphism from the Dolbeault resolution to the de Rham resolution of \({\mathcal O}_X\). complex manifolds; homotopic resolutions; de Rham complex de Rham cohomology and algebraic geometry, Analytic sheaves and cohomology groups, Homology and cohomology theories in algebraic topology, Resolutions; derived functors (category-theoretic aspects) Homotopic resolutions and the de Rham morphism
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The article under review is an expository one and discusses two basic cases of resolution of singularities. It starts with the existence of an equivariant resolution of singularities for a toric variety \(X_\Delta\) [\textit{G. Kempf, F. Knudsen, D. Mumford} and \textit{B. Saint-Donat}, Toroidal embeddings I, Lect. Notes in Mathematics 339. Berlin-Heidelberg-New York, Springer-Verlag (1973; Zbl 0271.14017)], proved by constructing a regular subdivision of the fan \(\Delta\). Next, a result of Khovansky [\textit{A. Varchenco}, Invent. Math. 37, 253--262 (1976; Zbl 0333.14007)] gives the existence of a weak resolution of singularities for a hypersurface \(H\subset\mathbb{C}^{n+1}\) defined by a non-degenerate irreducible polynomial \(f\). Its proof is based on subdividing the dual fan of the Newton polyhedron \(\Gamma_+(f)\) to obtain a regular subdivision of the cone \({\mathbb{R}}_{\geq 0}^{n+1}\). After introducing briefly canonical divisors and proving the adjunction formula, canonical, terminal, log-canonical and log-terminal singularities are defined. The last theorem gives a necessary and sufficient condition for a non-degenerate hypersurface \(H\) to have canonical or log-canonical singularities at 0 in terms of its Newton polyhedron \(\Gamma_+(f)\). Though the author works over the field of complex numbers, all results remain valid for an arbitrary algebraically closed field. Newton polygone; resolution of singularities; toric variety Ishii, S.; Brasselet, J-P (ed.); etal., A resolution of singularities of a toric variety and non-degenerate hypersurface, 354-369, (2007), Hackensack Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Hypersurfaces and algebraic geometry A resolution of singularities of a toric variety and a non-degenerate hypersurface
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Here the notion of a prehomogeneous vector space is generalized to that of a prehomogeneous affine space and the associated zeta functions are studied. More precisely, let \(\rho\) be an algebraic homomorphism of a linear algebraic group \(G\) into the affine transformation group \(\text{Aff} (V)\) of a finite dimensional vector space \(V\), and \(\alpha\) a regular rational function on \(V \times G\) satisfying the cocycle condition \(\alpha(x,gg')=\alpha(x \rho(g),g')+\alpha(x,g)\) \((x \in V,g,g' \in G)\). A quartet \({\mathbf D} = (G, V,\rho,\alpha)\) is called a prehomogeneous affine datum (PAD) if there exists a proper algebraic subset \(S\) of \(V\) such that \(V-S\) is a single \(\rho(G)\)-orbit. It is proved that the dual \({\mathbf D}^*=(G,V^*,\rho^*,\alpha^*)\) of \({\mathbf D}\) is a PAD if \({\mathbf D}\) is regular in a suitable sense.
Let \({\mathbf D}=(G,V,\rho,\alpha)\) be a regular PAD and \(L\) a lattice of \(V_ \mathbb{R}\). Let \(n\) be the number of independent relative invariants of \({\mathbf D}\). Then one can define the associated zeta functions \(\xi_ i(s,L)\) \((s \in \mathbb{C}^ n\), \(1 \leq i \leq \nu=\) the number of connected components of \(V_ \mathbb{R}-S_ \mathbb{R})\). Let \(\xi^*_ j(s,L^*)\) \((1 \leq j \leq \nu)\) be the zeta functions associated to the dual lattice \(L^*\) of \(L\). The main result of the paper is that, under certain assumptions on \({\mathbf D}\), the functional equations hold between \(\xi_ i (s,L)\) and \(\xi^*_ j(s,L^*)\). Hermitian form; prehomogeneous vector space; prehomogeneous affine space; zeta functions; functional equations Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Homogeneous spaces and generalizations Zeta functions of prehomogeneous affine spaces
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((B_0, \mathfrak{m}_{B_0})\) be a local ring, \(f : (A, \mathfrak{m}_A) \to (B, \mathfrak{m}_B)\) a deformation, i.e.~\(f\) is flat and \(B/\mathfrak{m}_B B \cong B_0\). Such a deformation is tangentially flat if the induced homomorphism \(\text{Gr}_{\mathfrak{m}_A} (A) \to \text{Gr}_{\mathfrak{m}_B} (B)\) of the associated graded rings is flat.
This notion is generalized to filtered local rings, \(f : (A, \mathfrak{m}_A, F_A) \to (B, \mathfrak{m}_B, F_B)\), \(F_A\) resp.~\(F_B\) a filtration on \(A\) resp.~\(B\), such that \(f(F^iA) \subseteq F^i_B\) and \(\text{Gr}_{F_A}(A) \to \text{Gr}_{F_B}(B)\) is flat. Several examples are computed and applications to Lech-Hironaka type inequalities \((H^{d+i}_A \leq H^i_B)\) between the Hilbert functions of the base and the total space are discussed. filtered local rings; Lech-Hironaka type inequalities; Hilbert functions Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Singularities of curves, local rings Local singularities, filtrations and tangential flatness
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system It is known that for each \(n\geq 3\) there exists a nonsingular algebraic curve C in \({\mathbb{R}}^ n\) such that the ideal \(I_ p(C)\) of \({\mathbb{R}}[x_ 1,...,x_ n]\) of polynomials vanishing on C cannot be generated by n-1 elements. Let \(O({\mathbb{R}}^ n)\) be the ring of regular functions on \({\mathbb{R}}^ n\). The ring \(O({\mathbb{R}}^ n)\) is naturally isomorphic to the localization of \({\mathbb{R}}[x_ 1,...,x_ n]\) with respect to the multiplicatively closed subset consisting of polynomials which do not vanish on \({\mathbb{R}}^ n\). It is shown in the paper that for any nonsingular algebraic curve C in \({\mathbb{R}}^ n\), the ideal I(C) of \(O({\mathbb{R}}^ n)\) of regular functions vanishing on C can be generated by n-1 elements. real algebraic curves as complete intersections DOI: 10.1007/BF01161973 Complete intersections, Real algebraic and real-analytic geometry, Special algebraic curves and curves of low genus Real algebraic curves as complete intersections
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We show that if \(X_n\) is a variety of \(c\times n\)-matrices that is stable under the group \(\operatorname{Sym}([n])\) of column permutations and if forgetting the last column maps \(X_n\) into \(X_{n-1} \), then the number of \(\operatorname{Sym}([n])\)-orbits on irreducible components of \(X_n\) is a quasipolynomial in \(n\) for all sufficiently large \(n\). To this end, we introduce the category of affine \(\mathbf{FI}^{op}\)-schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one \(\mathbf{FI}^{op} \)-scheme becomes of product form, where \(X_n=Y^n\) for some scheme \(Y\) in affine \(c\)-space. Furthermore, to any \(\mathbf{FI}^{op} \)-scheme of width one we associate a \textit{component functor} from the category \(\mathbf{FI}\) of finite sets with injections to the category \(\mathbf{PF}\) of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that \(\operatorname{Sym}([n])\)-orbits of components of \(X_n \), for all \(n\), correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem. We present applications of our methods to counting fixed-rank matrices with entries in a prescribed set and to counting linear codes over finite fields up to isomorphism. Components of symmetric wide-matrix varieties
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The purpose of this paper is to construct the algebraic and topological centralizer resolutions at the prime \(2\) and height \(2\), extending results that were announced in [\textit{H.-W. Henn}, Lond. Math. Soc. Lect. Note Ser. 342, 122--169 (2007; Zbl 1236.55015)]. The constructions of the related algebraic duality resolution and its topological counterpart (also announced in [loc. cit.]) have been given respectively in [\textit{A. Beaudry}, Algebr. Geom. Topol. 15, No. 6, 3653--3705 (2015; Zbl 1350.55019)] and [\textit{I. Bobkova} and \textit{P. G. Goerss}, J. Topol. 11, No. 4, 918--957 (2018; Zbl 1419.55014)], the latter using the centralizer resolutions. These are foundational results that have been important for recent progress in chromatic homotopy theory at the prime \(2\).
Let \(\Gamma\) be a formal group law of height \(2\) defined over \(\mathbb{F}_2\), here taken either to be the Honda law or the law associated to a supersingular elliptic curve over \(\mathbb{F}_2\). The Morava stabilizer group \(\mathbb{S}_2 (\Gamma)\) identifies as \(\mathrm{Aut}_{\mathbb{F}_4} (\Gamma)\) and is independent of the choice of \(\Gamma\); the extended Morava stablizer group \(\mathbb{G}_2 (\Gamma)\) is the semi-direct product \(\mathbb{S}_2 (\Gamma) \rtimes \mathrm{Gal}\) given by the Galois action, which depends upon \(\Gamma\).
The profinite group \(\mathbb{G}_2 (\Gamma)\) acts on \(\mathbb{W}\), the Witt vectors of \(\mathbb{F}_4\), via the Galois action. The main algebraic result is the construction of the algebraic centralizer resolution, which is an \(\mathcal{F}\)-resolution by \(\mathcal{F}\)-projective, Galois-twisted \(2\)-profinite \(\mathbb{G}_2 (\Gamma)\)-modules:
\[
0
\rightarrow
P_4
\rightarrow
P_3
\rightarrow
P_2
\rightarrow
P_1
\rightarrow
P_0
\rightarrow
\mathbb{W}
\rightarrow
0.
\]
In particular, each \(P_i\) is a finite direct sum of `induced' modules of the form \(\mathbb{W}\uparrow _G^{\mathbb{G}_2 (\Gamma)}\), for certain finite subgroups \(G \subset \mathbb{G}_2 (\Gamma)\), and the sequence is split exact when restricted to any finite subgroup.
The key input is the calculation of the mod \(2\) cohomology algebra of \(PS^1_2 : = S^1_2/ C_2\), where \(\mathbb{S}^1_2 (\Gamma)\subset \mathbb{S}(\Gamma)\) is defined using the reduced norm and \(S^1_2 (\Gamma)\) is its intersection with the normal \(2\)-Sylow subgroup of \(\mathbb{S}(\Gamma)\). This is analysed using Quillen's \(F\)-isomorphism theorem via restriction to elementary abelian \(2\)-groups; it is the origin of the terminology \textit{centralizer resolution}.
The author then constructs the topological centralizer resolution of the \(K(2)\)-local sphere \(L_{K(2)} S^0 \simeq E_2 ^{h \mathbb{G}_2 (\Gamma)}\), where \(E_2\) is the Lubin-Tate theory associated to \(\Gamma\). This realizes the algebraic centralizer resolution and has the form
\[
*
\rightarrow
L_{K(2)} S^0
\rightarrow
X_0
\rightarrow
X_1
\rightarrow
X_2
\rightarrow
X_3
\rightarrow
X_4
\rightarrow
*,
\]
where each \(X_i\) is a finite wedge of homotopy fixed point spectra of the form \(E_2 ^{hG}\), for certain finite subgroups \(G \subset \mathbb{G}_2 (\Gamma)\) as above. chromatic homotopy theory; Morava stabilizer group; centralizer resolution; topological resolution Stable homotopy theory, spectra, Bordism and cobordism theories and formal group laws in algebraic topology, Formal groups, \(p\)-divisible groups The centralizer resolution of the \(K(2)\)-local sphere at the prime \(2\)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this paper, a combination of algebraic and topological methods are applied to obtain new and structural results on harmonic rings. Especially, it is shown that if a Gelfand ring \(A\) modulo its Jacobson radical is a zero dimensional ring, then \(A\) is a clean ring. It is also proved that, for a given Gelfand ring \(A\), then the retraction map \(\mathrm{Spec}(A)\rightarrow\mathrm{Max}(A)\) is flat continuous if and only if \(A\) modulo its Jacobson radical is a zero dimensional ring. Dually, it is proved that for a given mp-ring \(A\), then the retraction map \(\mathrm{Spec}(A)\rightarrow\mathrm{Min}(A)\) is Zariski continuous if and only if \(\mathrm{Min}(A)\) is Zariski compact. New criteria for zero dimensional rings, mp-rings and Gelfand rings are given. The new notion of lessened ring is introduced and studied which generalizes ``reduced ring'' notion. Especially, a technical result is obtained which states that the product of a family of rings is a lessened ring if and only if each factor is a lessened ring. As another result in this spirit, the structure of locally lessened mp-rings is also characterized. Finally, it is characterized that a given ring \(A\) is a finite product of lessened quasi-prime rings if and only if \(\mathrm{Ker}\pi_{\mathfrak{p}}\) is a finitely generated and idempotent ideal for all \(\mathfrak{p}\in\mathrm{Min}(A)\). mp-ring; flat topology; Gelfand ring; lessened ring Relevant commutative algebra, Ideals and multiplicative ideal theory in commutative rings, General commutative ring theory, Injective and flat modules and ideals in commutative rings Structural results on harmonic rings and lessened rings
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system For simplicity, let \(V\) be a discrete valuation ring whose field of fractions has characteristic \(0\), although most results in the article under review are valid without the characteristic zero constraint.
The authors introduce and study the analytic cyclic homology of a certain (generally noncommutative) bornological \(V\)-algebras. For instance, the theory works for the Monsky-Washinitzer weak completion of a commutative, flat, finite type \(V\)-algebra.
The analytic cyclic homology of a bornological algebra \(A\) is a \(\mathbb{Z}/2\)-graded vector space over the field of fractions of \(V\), denoted by \(\mathrm{HA}_{\ast}(A)\).
The authors prove a number of desirable properties of \(\mathrm{HA}_{\ast}(A)\), such as the theory is dagger-homotopy invariant, satisfies excision, and is invariant under Morita equivalences. They also calculate \(\mathrm{HA}_{\ast}(A)\) explicitly when:
-- \(A\) is the Leavitt path algebra of a coutable graph,
-- \(A\) is the Cohn path algebra of a countable graph,
-- \(A\) is the Monsky-Washinitzer weak completion of a smooth relative affine curve over \(V\).
In the last case, it is proved that \(\mathrm{HA}_{*}(A)\) is isomorphic to the Monsky-Washinitzer cohomology (hence the rigid cohomology) of the special fiber. cyclic homology; dagger algebra; bornological algebra; Leavitt path algebra; excision; Cuntz-Quillen theory \(K\)-theory and homology; cyclic homology and cohomology, \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry, Rigid analytic geometry, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Leavitt path algebras Nonarchimedean analytic cyclic homology
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors consider a Poincaré resolution for the supersymmetric ring of brackets over a signed alphabet based on the general theory of resolutions developed in a paper by \textit{D. J. Anick} [Trans. Am. Math. Soc. 296, 641-659 (1986; Zbl 0598.16028)]. As a consequence, they outline a characteristic-free solution of the problem of computing the higher- order syzygies for the algebra of brackets or, what is equivalent, for the coordinate ring of the Grassmannian variety in projective space. The presented method does not require background in homological algebra. The authors give the syzygies of various order for the classical case of a bracket of length \(n=2\), over a negative alphabet. These brackets occur in symbolic methods for the expression of invariants of binary forms. Poincaré resolution; supersymmetric ring of brackets; characteristic- free solution; higher-order syzygies; coordinate ring; Grassmannian variety DOI: 10.1073/pnas.88.18.8087 ``Super'' (or ``skew'') structure, Vector and tensor algebra, theory of invariants, Grassmannians, Schubert varieties, flag manifolds, Resolutions; derived functors (category-theoretic aspects) Higher-order syzygies for the bracket algebra and for the ring of coordinates of the Grassmannian
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(C\) be a smooth connected projective curve. It is well-know that, given a semistable vector bundle \(V\) over \(C\), it admits a filtration
\[
0=V_0\subset V_1\subset\dots\subset V_k=V
\]
of vector subbundles, such that each quotient \(V_{i+1}/V_i\) is a stable vector bundle of the same slope as \(V\). The associated graded object is \(\mathrm{gr}(V)=\bigoplus_iV_{i+1}/V_i\). In case \(V\) is polystable, then \(\mathrm{gr}(V)\cong V\). The is the so-called Jordan-Hölder reduction for the vector bundle \(V\). In [\textit{A. Ramanathan}, Proc. Indian Acad. Sci., Math. Sci. 106, No.3, 301-328 (1996; Zbl 0901.14007)] it is proved the existence of a Jordan-Hölder reduction for every semistable principal \(G\)-bundle over the curve \(C\), where \(G\) is a reductive complex Lie group. If \(E\) is the bundle of linear frames of a rank \(n\) vector bundle \(V\), then the structure group of \(E\) is \(\mathrm{GL}(n,\mathbb{C})\), and giving a filtration of \(V\) is the same as giving a reduction of structure group of \(E\) to a parabolic subgroup \(P\subset\mathrm{GL}(n,\mathbb{C})\). In this case, a Jordan-Hölder reduction is then a certain reduction of structure group to a parabolic subgroup \(P\), such that the principal bundle obtained by extending the structure group to the Levi factor \(L(P)\) is stable. Later, \textit{C. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 75, 5--95 (1992; Zbl 0814.32003)] proved the existence of a Jordan-Hölder for semistable Higgs bundles. On the curve \(C\), a Higgs bundle is a pair \((V,\phi)\) where \(V\) is a vector bundle and \(\phi\) is a section of \(\mathrm{End}(V)\otimes\Omega^1\). In this case, a Jordan-Hölder reduction is a filtration of \(E\) as above but such that each \(V_i\) is \(\phi\)-invariant i.e. \(\phi(V_i)\subset V_i\otimes\Omega^1\).
This main result of this paper is the proof of the existence of a Jordan-Hölder reduction for every semistable Higgs \(G\)-bundle over \(C\), where \(G\) is a complex reductive Lie group. Over \(C\), a Higgs \(G\)-bundle is a pair \((E,\phi)\) where \(E\) is a principal \(G\)-bundle and \(\phi\) is global section of \(\mathrm{Ad}(E)\otimes\Omega^1\), where \(\mathrm{Ad}(E)\) is the adjoint bundle. In the same spirit of Ramanathan's work, a Jordan-Hölder reduction of a Higgs \(G\)-bundle \((E,\phi)\) is a certain reduction of structure group of \((E,\phi)\) (hence compatible with \(\phi\)) to a parabolic subgroup \(P\subset G\), such that the Higgs \(G\)-bundle obtained by extending the structure group to the Levi factor \(L(P)\) of \(P\) is a stable principal Higgs \(L(P)\)-bundle. principal Higgs bundles; semistability; Jordan-Hölder Graña~Otero, B., Jordan-Hölder reductions for principal Higgs bundles on curves, J. geom. phys., 60, 11, 1852-1859, (2010) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on curves and their moduli Jordan-Hölder reductions for principal Higgs bundles on curves
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(V_0,\ldots,V_m\) be finite dimensional vector spaces over an algebraically closed field, \(L_m\) the associated variety of complexes with an action of the product of linear groups on those spaces.
The author uses the methods of \(F\)-splitting and Kempf's resolution to show that the orbit closure \( \overline{O}_{f_m}\) for \(f_m\in L_m\) is normal, Cohen-Macaulay and with rational singularities.
This result was proved by means of different methods by \textit{G. Kempf} [Bull. Am. Math. Soc. 81, 900-901 (1975; Zbl 0322.14020)] and \textit{C. Musili} and \textit{C. S. Seshadri} [in: Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 329-359 (1983; Zbl 0567.14030)] and others. Cohen-Macaulay variety; \(F\)-splitting; Kempf's resolution; action of linear groups; Schubert variety; variety of complexes V.B. Mehta and V. Trivedi, Variety of complexes and \(F\) -splitting, J. Alg. 215 (1999), 352-365. Group actions on varieties or schemes (quotients), Syzygies, resolutions, complexes and commutative rings, Singularities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Variety of complexes and \(F\)-splitting
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be a field of characteristic zero and let \(R\) be the polynomial ring in \(n\) variables over \(K\).
An artinian graded \(K\)-algebra \(R/I\) is said to have the Strong Lefschetz Property (SLP) whenever, for a general linear form \(L\) and for any integer \(d\), the multiplication map given by \(L^d\) from \([R/I]_t\to [R/I]_{t+d}\) has maximal rank in every degree \(t\). If this happens for \(d=1\), then \(R/I\) is said to have the Weak Lefschetz Property (WLP).
Understanding whenever \(R/I\) has the SLP and the WLP is a challenging and motivating problem that has inspired a lot of papers and it has generated an incredible amount of connections between different areas of mathemathics.
A general classification of the WLP and the SLP is not known, but many special cases have been invastigated. In particular, the case when the ideal \(I\) is generated by powers of linear forms is very interesting. In this setting the failure of the WLP is related, among all, to the problem wheter a set of fat points imposes the expected number of conditions on a linear systems of forms of fixed degree.
When the number of variables is \(n=3\), it is known from [\textit{H. Schenck} and \textit{A. Seceleanu}, Proc. Am. Math. Soc. 138, No. 7, 2335--2339 (2010; Zbl 1192.13013)] that \(R/I\) has the WLP. In this paper the authors extend such result by showing that the multiplication by \(L^2\) always has maximal rank and they characterize whenever the multiplication by \(L^3\) has maximal rank. These results are proved in Section 5 and 6 and summarized in Theorem 1.1.
Many consequences of Theorem 1.1 apply to algebras in 4 variables. In particular in sections 6 and 7 are examinated the cases when either \(I\) is generated in one degree (see Corollary 6.5. ) or the initial degree of \(I\) is low (2 or 3) see Theorem 1.2. These results are derived by a more general and powerful result based on an ``exchange property'', see proposition 3.1, which connects the Lefschetz properties of different rings and allows inductive approaches based on the number of variables and generators of \(I\). Cremona transformation; inverse system; maximal rank Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Commutative Artinian rings and modules, finite-dimensional algebras, Projective techniques in algebraic geometry The Lefschetz question for ideals generated by powers of linear forms in few variables
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Consider a blow-up square of schemes of the form
\[
\begin{tikzcd} Y \arrow[r] \arrow[d] & X\arrow[d, "f"]\\ V \arrow[r] & \mathrm{Spec}(A)\end{tikzcd}
\]
Fix a non-negative integer \(n\). Motivated by analogies with Gersten's conjecture, the authors seek conditions under which the map
\[
K_n(A,I)\rightarrow K_n(A/I^r,I/I^r)\oplus K_n(X,Y_{\text{red}})\eqno(1)
\]
is injective for all sufficiently large \(r\). When \(A\) is local, noetherian, quasi-excellent and contains a field of characteristic \(0\), the authors establish injectivity if \(Y_{\text{red}}\) is regular. (For \(n=2\), this last condition can be dropped.)
It follows from this and some additional argument that if \(k\) is a field of characteristic \(0\) and \(C\hookrightarrow {\mathbb A}^{N+1}_k\) is the cone over a smooth projective variety with \((A,{ m})\) the local ring at the singular point, then we have injectivity of
\[
K_n(A)\rightarrow K_n(A/m^r)\oplus K_n(Spec(A)-\{m\})\eqno(2)
\]
again for sufficiently large \(r\).
The authors show by counterexample that (2) need not be injective for general isolated singularities, even in dimension one. They conjecture, however, that (1) holds in much greater generality.
The proofs rely heavily on computations in cyclic and Hochschild homology. Gersten's conjecture; algebraic \(K\)-theory; singular schemes; Hochschild homology; cyclic homology \(K\)-theory of schemes, Singularities in algebraic geometry Analogues of Gersten's conjecture for singular schemes
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\mathcal A_g\) be the moduli stack of \(g\)-dimensional principally polarized abelian varieties. In this paper, the author studies the boundary contribution of the cohomology of a local system on \(\mathcal A_g\), viewed as either a mixed Hodge structure for the complex topology, or as a Galois representation for the etale topology. For each dominant weight \(\lambda=(\lambda_1,\dots, \lambda_g)\) with \(\lambda_1\geq \lambda_2 \geq \dots \geq \lambda_g\) of \(\text{Gsp}_{2g}\), one associates a local system \(\mathbb V_\lambda\) on \(\mathcal A_g\). The cohomology of this local system is closely related to vector-valued Siegel modular forms. Consider the Euler characteristic \( e(\mathcal A_g,\mathbb V_\lambda):=\sum_{i} (-1)^i [H^i(\mathcal A_g, \mathbb V_\lambda)] \) in the Grothendieck group of an appropriate category, i.e. MHS or Galois representations. Similarly consider the compactly supported analogue \( e_c(\mathcal A_g,\mathbb V_\lambda):=\sum_{i} (-1)^i [H^i_c(\mathcal A_g, \mathbb V_\lambda)]. \) The Euler characteristic of the Eisenstein cohomology is defined as the difference \( e_{\text{Eis}}(\mathcal A_g,\mathbb V_\lambda):= e(\mathcal A_g,\mathbb V_\lambda)-e_c(\mathcal A_g,\mathbb V_\lambda). \) Let \(\tilde {\mathcal A}_g\) be a smooth toroidal compactification of \(\mathcal A_g\). The Satake compactification \(\mathcal A^*_g\) has a natural stratification \(\mathcal A^*_g=\coprod_{i=0}^g \mathcal A_i\). Let \(q:\tilde {\mathcal A}_g \to \mathcal A^*_g\) and \(j:{\mathcal A}_g \to \tilde {\mathcal A}_g\) be the natural maps. One can rewrite \(e_{\text{Eis}}\) as \( e_{\text{Eis}}(\mathcal A_g,\mathbb V_\lambda)=e(\tilde {\mathcal A}_g, Rj_* \mathbb V_\lambda)-e(\tilde {\mathcal A}_g, Rj_! \mathbb V_\lambda)= \sum_{i=1}^{g-1} e_c( q^{-1}(\mathcal A_i) ,Rj_* \mathbb V_\lambda- Rj_! \mathbb V_\lambda). \) The contribution of the rank 1 part is defined as \( e_{\text{Eis,1}}(\mathcal A_g,\mathbb V_\lambda):= e_c(q^{-1}(\mathcal A_{g-1}) ,Rj_* \mathbb V_\lambda- Rj_! \mathbb V_\lambda). \) The main theorem (Theorem 2.1) computes \(e_{\text{Eis,1}}\) in terms of compactly supported cohomology of another explicit local system on \(\mathcal A_{g-1}\). The author also computes the full \(e_{\text{Eis}}\) for \(g=1,2\). Eisenstein cohomology; Siegel modular varieties; BGG complex Geer, G, Rank one Eisenstein cohomology of local systems on the moduli space of abelian varieties, Sci. China Math., 54, 1621-1634, (2011) Algebraic moduli of abelian varieties, classification, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Hecke-Petersson operators, differential operators (several variables), Modular and Shimura varieties, Moduli, classification: analytic theory; relations with modular forms Rank one Eisenstein cohomology of local systems on the moduli space of abelian varieties
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The group \(G=\mathrm{GL}_r(k)\times (k^\times)^n\) acts on \(\mathbf{A}^{r\times n}\), the space of \(r\)-by-\(n\) matrices: \(\mathrm{GL}_r(k)\) acts by row operations and \((k^\times)^n\) scales columns. A matrix orbit closure is the Zariski closure of a point orbit for this action. We prove that the class of such an orbit closure in \(G\)-equivariant \(K\)-theory of \(\mathbf{A}^{r\times n}\) is determined by the matroid of a generic point. We present two formulas for this class. The key to the proof is to show that matrix orbit closures have rational singularities. Equivariant \(K\)-theory classes of matrix orbit closures
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We give an improved version of the Briançon-Skoda theorem for regular rings containing a field. If \((R,m)\) is such a ring, \(I\) is an ideal of analytic spread \(\ell\) and bigheight \(h\), and \(J\) is a reduction of \(I\) then we show that \(I^ \ell \subseteq J(I^{\ell-h})^{\text{un}}\) (where \(( )^{\text{un}}\) means: take the intersection of the minimal primary components). The proof uses tight closure in characteristic \(p\) and then standard techniques of reduction to characteristic \(p\) for local rings containing fields of characteristic 0.
We then apply this result to study the relationship between the Cohen- Macaulayness of \(R[It]\) and \(\text{Gr}_ I(R)\). In particular we show that if \(R\) is a regular local ring containing a field and \(I\) is an unmixed curve then \(R[It]\) is Cohen-Macaulay if and only if \(\text{Gr}_ I(R)\) is Cohen-Macaulay.
In a final section we continue the investigation of 4-generated unmixed ideals of height 2 in a 3-dimensional regular local ring begun by Vasconcelos. characteristic \(p\); Briançon-Skoda theorem; regular rings; analytic spread; tight closure; Cohen-Macaulayness; 4-generated unmixed ideals Aberbach I. and Huneke, C. : An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees rings , Math. Ann. 297 (1993) 343-369. Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Cohen-Macaulay modules, Regular local rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) An improved Briançon-Skoda theorem with applications to the Cohen- Macaulayness of Rees algebras
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system For a commutative ring \(S\), \(S^{[n]}\) denotes a polynomial ring in \(n\) variables over \(S\). For a prime ideal \(P\) of \(S\), \(k(P)\) denotes the field \(SP/PSP\).
An \(S\)-algebra \(A\) is said to be an affine \(n\)-fibration over \(S\) (denoted by \(\mathbb{A}^n)\), if the following conditions hold:
(i) \(A\) is a finitely generated flat \(S\)-algebra,
(ii) For every prime ideal \(P\) of \(S\), \(A\otimes_S k(P)= k(P)^{[n]}\).
In this survey article the authors highlight some recent developments in the problems on affine fibrations. They show how some recent results completely determine the structure of an \(\mathbb{A}^1\) fibration over a seminormal ring and how these results can be used to prove a generalised epimorphism theorem. It is also shown how the problem of \(\mathbb{A}^2\) fibration over a two-dimensional regular local ring is related to the problem of embedding of a plane in affine 3-space over a discrete valuation ring. affine fibrations S. M. Bhatwadekar and A. K. Dutta, \textit{On affine fibrations, Commutative Algebra} (ed. A. Simis, N.V. Trung, G. Valla), World Sc. (1994) 1-17. Polynomial rings and ideals; rings of integer-valued polynomials, Fibrations, degenerations in algebraic geometry On affine fibrations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let R denote an n-dimensional local ring with maximal ideal m. The purpose of the paper is to show the following two theorems which are related to weak intersection conjecture, and canonical element conjecture, respectively.
Theorem 1. Assume depth R\(=k<n\) and \(H^ k_ m(R)\) is decomposable. Then every non zero R-module of finite length has an infinite projective dimension. (A partial solution to weak intersection conjecture.) - Theorem 2. Let \(x_ 1,...,x_ n\) be a system of parameters for R, K(t)\(\bullet\) the Koszul complex with respect to \(x^ t_ 1,...,x^ t_ n\), G(t)\(\bullet\) a free resolution of \(R/(x^ t_ 1,...,x_ n^ t)\) and g(t)\(\bullet\) a map from K(t)\(\bullet\) to G(t)\(\bullet\) which lifts the identity map on \(R/(x^ t_ 1,...,x^ t_ n)\). Assume depth R\(=n-1\) and \(H_ m^{n-1}(R)\) is of finite length. Then, for all sufficiently large t, \(g(t)_ i\) splits for every \(i\neq n\). (Canonical element conjecture asserts \(g(t)_ n\) splits.) - The proofs are done by studying Koszul homology modules, Tor modules and some exact sequences. n-dimensional local ring; weak intersection conjecture; canonical element conjecture; Koszul homology; Tor Jee, R., 1985. A study on projection pursuit methods. Unpublished Ph.D. Thesis, Rice University, USA. (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Local rings and semilocal rings, Local cohomology and algebraic geometry A note on the first non-vanishing local cohomology module
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\mathbb{P}^ n\) be the projective space of dimension \(n\) over an algebraically closed field of characteristic zero. A finite subscheme \(X\) in \(\mathbb{P}^ n\) is said to be in linearly general position if for every proper linear subspace \(L \subset \mathbb{P}^ n\) one has \(\deg (X \cap L) \leq 1 + \dim (L)\). If one cuts a reduced, irreducible, nondegenerate projective variety \(V\) of codimension \(n\) in \(\mathbb{P}^ r\) with a general linear subspace of dimension \(n\), one gets a set \(X\) of distinct points on this \(\mathbb{P}^ n\) in linear general position. This explains why this notion has a central role in many problems of algebraic geometry, especially in the so-called Castelnuovo theory as developed by Castelnuovo and more recently by \textit{J. Harris} (with the collaboration of \textit{D. Eisenbud}) [``Curves in projective space'', Sém. Math. Supér. 85 (1982; Zbl 0511.14014)] and \textit{M. L. Green} [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)]. A classical result in this field is the so-called Castelnuovo lemma which states that any set \(X \subset \mathbb{P}^ n\) of \(d \geq 2n + 3\) points in linearly general position, which imposes a most \(2n + 1\) conditions on the system of quadrics in \(\mathbb{P}^ n\), lies on a rational normal curve. In the quoted paper, M. Green showed a more subtle result, the so-called strong Castelnuovo lemma, for any set \(X \subset \mathbb{P}^ n\) of points in linearly general position.
Continuing recent works of D. Bayer, D. Eisenbud and J. Harris, leading to extend Castelnuovo theory to more general finite subschemes of the projective space, in the present paper the authors extend the proof of the strong Castelnuovo lemma for zero-dimensional schemes in linearly general position from the reduced to the nonreduced case. linearly general position; strong Castelnuovo lemma; zero-dimensional schemes Cavaliere, M. P.; Rossi, M. E.; Valla, G.: The strong Castelnuovo lemma for zerodimensional schemes. (1994) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves, Schemes and morphisms, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The strong Castelnuovo lemma for zerodimensional schemes
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In the article under review the author considers the affine variety \(\Hom(V\otimes V,V)\), where \(V\) is an \(n\)-dimensional vector space over the field of complex numbers. There is an algebraic subset \(\text{Leib}_n({\mathbb{C}})\subset \Hom(V\otimes V,V)\) which consists of elements \(\lambda\) such that the pair \((V,\lambda)\) is a Leibniz algebra. Similarly \(\text{LN}_n(\mathbb{C})\) is a subset consisting of elements \(\lambda\) such that the pair \((V,\lambda)\) is a nilpotent Leibniz algebra. Some closed (in Zariski topology) subsets of \(\text{LN}_n(\mathbb{C})\) are described in the article. There is an action of \(\text{GL}_n(\mathbb{C})\) on \(\text{Leib}_n({\mathbb{C}})\). For given elements \(\lambda,\mu\in \text{Leib}_n({\mathbb{C}})\) one can say that \(\lambda\) degenerates to \(\mu\) if \(\mu\) lies in the Zariski closure of the orbit of \(\lambda\). The author proves some statements about degenerations as well. affine variety; Zariski topology DOI: 10.1007/s10958-006-0218-3 Affine geometry, Leibniz algebras On the degenerations of finite dimensional nilpotent complex Leibniz algebras
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The paper contains numerous important theorems and methods concerning some old and new homological conjectures. The author's introduction begins with: ''One of the objectives of this paper is to show that the usual homological consequences of the existence of big Cohen-Macaulay modules follow from the direct summand conjecture when the residual characteristic of the local ring is positive.'' At the same time he develops ''a theory of certain canonical elements in the local cohomology of special modules of syzygies'': Let (R,\({\mathfrak m},K)\) be a local ring of dimension d and consider an exact R-sequence \({\mathcal E}:0\to S\to P_{d- 1}\to...\to P_ 1\to R\to K\to 0\) with finitely generated free modules \(P_ i\). The sequence \({\mathcal E}\) represents an element of \(Ext^ d_ R(K,S)\) which is mapped to an element \(\eta_ R=\eta_ R({\mathcal E})\) of \(H^ d_{{\mathfrak m}}(S)\) by the natural transformation of functors \(Ext^*_ R(K,\quad)\to H^*_{{\mathfrak m}}(\quad).\) \(\eta_ R\) is called the ''canonical element'' in \(H^ d_{{\mathfrak m}}(S)\). One can identify the various \(\eta_ R's\) obtained from various \({\mathcal E}'s\). Furthermore \(\eta_ R\neq 0\) can be shown to be equivalent to the following property: (CE) For every projective resolution \(P_*:...\to P_ i\to...\to P_ 0\to K\to 0\) and for every system of parameters \(\underline x=x_ 1,...,x_ d\) for R, if \(\phi\) is any homomorphism of complexes from the usual Koszul complex \(K_*(\underline x;R)\) to \(P_*\) which lifts the natural projection R/\b{x}R\(\to K\), then \(\phi_ d: K_ d(\underline x;R)(=R)\to P_ d\) is nonzero.
Some of the main results about the ''usual'' homological conjectures and property (CE): (1) If R satisfies (CE), then the new intersection conjecture of Peskine-Szpiro and Roberts is true for R. - (2) If R is a Cohen-Macaulay domain such that all local rings of homomorphic image domains of R satisfy (CE), then the syzygy problem of Evans-Griffith can be settled affirmatively for R. - (1) and (2) are in some sense improvements of results if \textit{E. G. Evans} jun. and \textit{P. Griffith} [Ann. Math., II. Ser. 114, 323-333 (1981; Zbl 0497.13013)] because: (3) If R has a big Cohen-Macaulay module, then R satisfies (CE). - With respect to these theorems the following result seems the most important one: (4) If the direct summand conjecture is true, then every local ring satisfies (CE). The converse of (4) also holds which gives property (CE) the maturity to become a conjecture.
Canonical element conjecture: For every local ring R, \(\eta_ R\neq 0\). Of course \(\eta_ R\neq 0\) is true in case the residual characteristic of R is 0 since in that case R is known to have a big Cohen-Macaulay module. When R has positive prime characteristic the author gives two proofs of \(\eta_ R\) being nonzero, one based on the functorial behavior of \(\eta_ R\) and the other - more complicated - one by establishing property (CE). The author then studies the connection of property (CE) with canonical modules and, in a separate section, the behaviour of \(\eta\) passing from R to a quotient of R by a nonzero-divisor. - The last section of the paper is devoted to a detailed investigation of the direct summand conjecture. The main theorem contains a reduction of the general case to the case of a formal power series ring over a complete unramified discrete valuation ring. Part of the ideas used to prove the theorem gives a new proof of the conjecture in the case of positive characteristic. property CE; local cohomology; modules of syzygies; intersection conjecture; syzygy problem; direct summand conjecture; Canonical element conjecture; canonical modules; formal power series ring [H] Hochster, M.: Canonical elements in local cohomology modules and the direct summand conjecture. J. Algebra84, 503--53 (1983) (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Deformations and infinitesimal methods in commutative ring theory, Local deformation theory, Artin approximation, etc., Complexes Canonical elements in local cohomology modules and the direct summand conjecture
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We study the geometry and the singularities of the principal direction of the Drinfeld-Lafforgue-Vinberg degeneration of the moduli space of \(G\)-bundles \(\mathrm{Bun}_G\) for an arbitrary reductive group \(G\), and their relationship to the Langlands dual group \(\check{G}\) of \(G\).
The article consists of two parts. In the first and main part, we study the monodromy action on the nearby cycles sheaf along the principal degeneration of \(\mathrm{Bun}_G\) and relate it to the Langlands dual group \(\check G\). We describe the weight-monodromy filtration on the nearby cycles and generalize the results of [37] from the case \(G=\mathrm{SL}_2\) to the case of an arbitrary reductive group \(G\). Our description is given in terms of the combinatorics of the Langlands dual group \(\check G\) and generalizations of the Picard-Lefschetz oscillators found in [37]. Our proofs in the first part use certain local models for the principal degeneration of \(\mathrm{Bun}_G\) whose geometry is studied in the second part.
Our local models simultaneously provide two types of degenerations of the Zastava spaces; these degenerations are of very different nature, and together equip the Zastava spaces with the geometric analog of a Hopf algebra structure. The first degeneration corresponds to the usual Beilinson-Drinfeld fusion of divisors on the curve. The second degeneration is new and corresponds to what we call \textit{Vinberg fusion}: it is obtained not by degenerating divisors on the curve, but by degenerating the group \(G\) via the Vinberg semigroup. Furthermore, on the level of cohomology the degeneration corresponding to the Vinberg fusion gives rise to an algebra structure, while the degeneration corresponding to the Beilinson-Drinfeld fusion gives rise to a coalgebra structure; the compatibility between the two degenerations yields the Hopf algebra axiom. geometric representation theory; geometric Langlands program; moduli spaces of \(G\)-bundles; nearby cycles; Picard-Lefschetz theory; weight-monodromy theory; Vinberg semigroup; Langlands duality Geometric Langlands program (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems, Langlands-Weil conjectures, nonabelian class field theory, Vector bundles on curves and their moduli, Fibrations, degenerations in algebraic geometry Monodromy and Vinberg fusion for the principal degeneration of the space of \(G\)-bundles
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((G_n)_{n\geq 0}\) be a projective system of finite groups, with surjective morphisms \(s_n:G_n\rightarrow G_{n-1}.\) To such a system of groups, a tower \((H_n)_{n\geq 0}\) of algebraic varieties is attached, such that \(H_n\) is geometrically irreducible and defined over some cyclotomic extension of \(\mathbb Q\). Moreover, the system \((H_n)\) has the following properties: each \(H_n\) is a component of some moduli space of Galois covers of the group \(G_n\); there exist projective systems of \({\mathbb Q}^{ur}_p\)-points, for all \(p,\) such that all \(G_n\) are \(p'\)-groups; there exist projective systems of \({\mathbb Q}^{ab}((x))\)-points and there exist projective systems of \textbf{R}-points.
This was the original motivation of the paper. The author's idea is to use the Harbater-Mumford components of Hurwitz spaces. The main fact proved in the paper is that the \(p\)-adic covers constructed by Harbater's patching methods lie on the Harbater-Mumford components. regular inverse Galois problem; Hurwitz moduli spaces; algebraic cover Dèbes, P., Emsalem, M.: Harbater-Mumford Components and Towers of Moduli Spaces. J. Inst. Math. Jussieu, to appear. Inverse Galois theory, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Families, moduli of curves (algebraic), Deformations of complex structures Harbater-Mumford components and towers of moduli spaces
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(k\) be a field of characteristic zero, and let \(I\) be an ideal in the polynomial ring \(R=k[x_1,\ldots,x_n]\). Denote by \({\mathbf m}_\zeta=(x_1-\zeta_1,\ldots,x_n-\zeta_n)\) the maximal ideal of \(R\) corresponding to the closed point \(\zeta\in\text{Spec} R\). Using basic properties of the Macaulay inverse system associated to \(I\) [see \textit{F. S. Macaulay}, ``The algebraic theory of modular systems'' (1916; JFM 46.0167.01); see also the reprint (1994; Zbl 0802.13001)] the author proposes an effective algorithm which enables one to describe the \({\mathbf m}_\zeta\)-primary component of \(I\). He then explains how this algorithm can be applied to the computation of local residues, to the analysis of real branches of a curve which is locally complete intersection, and to the computation of resultants of homogeneous polynomials. Macaulay inverse system; primary ideal; primitive ideal; local residue; complete intersection curve; Gröbner basis; polynomial ideal Mourrain, B.: Isolated points, duality and residues. J. pure appl. Algebra 117 \& 118, 469-493 Special issue for the Proceedings of the 4th Int. Symp. on Effective Methods in Algebraic Geometry (MEGA) (1996) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Ideals and multiplicative ideal theory in commutative rings, Effectivity, complexity and computational aspects of algebraic geometry, Polynomials over commutative rings Isolated points, duality and residues
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R, \mathfrak m)\) be a 2-dimensional regular local ring with an algebraically closed coefficient field. A divisorial valuation \(\nu\) is a discrete valuation of the fraction field of \(R\), centered at \((R, \mathfrak m)\). In fact there is a 1-1 correspondance between divisorial valuations and finite sequences of blowing-ups. Let \(V\) be a finite set \(\{\nu_1,\dots,\nu_r\}\) of divisorial valuations centered at a 2-dimensional regular local ring \(R\).
In this paper the authors study its structure by means of the semigroup of values, \(S_V=\{\nu_1(f),\dots,\nu_r(f), f\in R\setminus \{0\}\}\subset \mathbb Z^r\), and the multi-index graded algebra defined by \(V\), gr\(_V\) \(R\). They prove that \(S_V\) is finitely generated and they compute its minimal set of generators following the study of reduced curve singularities. Moreover, they prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in \(V\), the approximation of a reduced plane curve singularity \(C\) by families of sets \(V^{(k)}\) of divisorial valuations, and the relationship between the value semigroup of \(C\) and the semigroups of the sets \(V^{(k)}\), allow the authors to obtain the (finite) minimal generating sequences for \(C\) as well as for \(V\).
They also analyze the structure of the homogeneous components of gr\(_V\) \(R\). The study of their dimensions allows the authors to relate the Poincaré series for \(V\) and for a general curve \(C\) of \(V\). Since the last series coincides with the Alexander polynomial of the singularity, they can deduce a formula of A'Campo type for the Poincaré series of \(V\). Moreover, the Poincaré series of \(C\) could be seen as the limit of the series of \(V^{(k)}, k\geqslant 0\). discrete valuation; divisorial valuation; blow-up; regular surface; reduced plane curves; semigroup of valuations; Poincaré series La Mata, F. Delgado-De; Galindo, C.; Núñez, A.: Generating sequences and Poincaré series for a finite set of plane divisorial valuations. Adv. math. 219, No. 5, 1632-1655 (2008) Singularities in algebraic geometry, Graded rings and modules (associative rings and algebras), Filtered associative rings; filtrational and graded techniques, Valuations and their generalizations for commutative rings Generating sequences and Poincaré series for a finite set of plane divisorial valuations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This is an announcement of a description of loop Grassmannians of reductive groups in the setting of ``local spaces'' over a curve. The structure of a local space is a part of the fundamental structure of a factorisable space introduced and studied by \textit{A. Beilinson} and \textit{V. Drinfeld} [Chiral algebras. Colloquium Publications. American Mathematical Society 51. Providence, RI: American Mathematical Society, 375 p. (2004; Zbl 1138.17300)]. The weakening of the requirements formalizes some well-known examples of ``almost factorisable'' spaces and constructions with such spaces.
The main observation of the paper is that the point of view of local spaces produces a generalization of loop Grassmannians corresponding to central extensions of loop groups of tori. The last section advertises local spaces as a setting for the conjecture of \textit{P. Baumann} and \textit{J. Kamnitzer} [Represent. Theory 16, 152--188 (2012; Zbl 1242.05273)] and Allen Knutson on a topological reconstruction of certain pieces of the loop Grassmannian (the MV-cycles) in terms of representations of quivers. Most of the content in this paper comes from a joint work with Kamnitzer and Baumann (loc. cit.) and Knutson. Grassmannians, Schubert varieties, flag manifolds, Loop groups and related constructions, group-theoretic treatment, Families, moduli of curves (algebraic) Loop Grassmannians in the framework of local spaces over a curve
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\Phi : F \to G^*\) be a generic map of locally free modules of rank \(m = \text{rk} F \geq n = \text{rk} G\) over a scheme \(X\) of the form \(X = \text{Spec(Sym} (F_ 0 \otimes_{{\mathcal O}_{X_ 0}} G_ 0))\), where \(X_ 0\) is a scheme defined over \(\mathbb{Q}\). For any partition \(\lambda\) there is an induced map \(\Phi_ \lambda : L_ \lambda F \to L_ \lambda G^*\) of Schur functors. The author proves that if \(\lambda\) is a rectangular partition, the homological dimension of the module \(M_ \lambda = \text{coker} (\Phi_ \lambda)\) is \(D(\lambda) (m - n) + 1\), where \(D(\lambda)\) is the size of the Durfee square of \(\lambda\), i.e. the largest square partition contained in \(\lambda\). This generalizes results of \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Adv. Math. 18, 245-301 (1975; Zbl 0336.13007)] and of \textit{K. Akin}, \textit{D. A. Buchsbaum} and \textit{J. Weyman} [Adv. Math. 39, 1-30 (1981; Zbl 0474.14035)]. More precisely, the author determines the syzygy modules of \(M_ \lambda\) in the case \(\lambda = (r^ s)\) by pushing down a certain Schur complex on the Grassmannian \(Y = \text{Grass}_{n - r} (G)\) and by using the spectral sequences of hypercohomology and Bott's theorem to calculate the result. Schur functors; syzygy modules; Grassmannian; hypercohomology Syzygies, resolutions, complexes and commutative rings, Determinantal varieties, Cohen-Macaulay modules Syzygies of a certain family of generically imperfect modules
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system For a partition \(\lambda=(\lambda_1,\dots,\lambda_m)\) of \(n\) denote by \(V_{\mathbb{R}}(\lambda)\) the Zariski closure in the space of real symmetric \(n\times n\) matrices of the subset of matrices whose eigenvalue multiplicities are given by \(\lambda\). This real algebraic variety is studied in the paper. Using the Cayley transform the author first gives a rational parametrization of this variety. It is used to compute the dimension of the complexification of \(V_{\mathbb{R}}(\lambda)\). For \(n=3\) and \(4\), explicit equations defining the varieties \(V_{\mathbb{R}}(\lambda)\) are found with the aid of computer. The specialization map to diagonal matrices restricts to an isomorphism between the subspace of orthogonal invariants in the vanishing ideal of \(V_{\mathbb{R}}(\lambda)\) and the subspace of \(S_n\)-invariants in the vanishing ideal of the variety \(V_{\mathbb{R}}(D_{\lambda})\) of diagonal matrices with eigenvalue multiplicities given by \(\lambda\). It is pointed out that the degree of \(V_{\mathbb{R}}(D_{\lambda})\) coincides with the Euclidean distance degree of \(V_{\mathbb{R}}(\lambda)\). real algebraic variety; real symmetric matrices; Euclidean distance degree; degenerate matrices Computational real algebraic geometry, Real algebraic sets, Eigenvalues, singular values, and eigenvectors, Hermitian, skew-Hermitian, and related matrices Real symmetric matrices with partitioned eigenvalues
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this paper, we give a simple proof of a famous result of \textit{O. Zariski} [Ann. Math. (2) 76, 560--615 (1962; Zbl 0124.37001)] and of \textit{T. Fujita} [J. Fac. Sci. Univ. Tokyo Sect. I A Math. 30, No. 2, 353--378; (1983; Zbl 0561.32012)] on linear systems with removable base loci. The novelty of our proof is that it is based on studying the canonical nonexact Koszul complex associated with the linear system. Ein, L., Linear systems with removable base loci, Commun. Algebra, 28, 5931-5934, (2000) Divisors, linear systems, invertible sheaves Linear systems with removable base loci.
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(k\) be a field of characteristic zero, \(L/k\) an \(n\)-dimensional algebraic function field, \(K\) a finite algebraic extension of \(L\), \(\nu\) a zero-dimensional valuation of \(K/k\), and \((R,M)\) a regular local ring, essentially of finite type over \(k\) with quotient field \(K\) and ground field \(k\) such that \(\nu\) has center \(M\) in \(R\). Then for some sequence of monoidal transforms \(R\to R^*\) along \(\nu\), there exists a local domain \(S^*\), essentially of finite type over \(k\) with quotient field \(L\) and ground field \(k\) lying below \(R^*\). When \(n=2\) this is stated by \textit{S. S. Abhyankar} in theorem 4.8 of his book ``Ramification theoretic methods in algebraic geometry'' [Ann. Math. Stud. No. 43 (1959; Zbl 0101.38201)]. The above result is a kind of simultaneous resolution of singularities; other forms are also included. resolution of singularity; birational domination; algebraic function field; valuation; regular local ring; monoidal transforms Cutkosky, S. D.: Simultaneous resolution of singularities. Proc. amer. Math. soc. 128, 1905-1910 (2000) Global theory and resolution of singularities (algebro-geometric aspects), Valuations and their generalizations for commutative rings, Singularities in algebraic geometry Simultaneous resolution of singularities
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be an algebraically closed field and \(X\) a reduced connected scheme of finite type over \(K\). In the paper under review, it is shown that for a proper map \(f:\;X\rightarrow {\mathbb P}^n\), the space \(H^0(X,f^*(T_{{\mathbb P}^n}(-1)))\) is \(n+1\)-dimensional, under the assumption that the image under \(f\) of each irreducible component of \(X\) is at least \(2\)-dimensional, and that the image of the intersection of two components is either empty or at least \(1\)-dimensional. By work of \textit{G. Hein} [Rocky Mt. J. Math. 30, No. 1, 217--235 (2000; Zbl 0983.14011)] it is known that this cannot be generalized to maps with \(1\)-dimensional image.
Via Zariski's main theorem, this result is used to address a problem motivated by the study of intersection multiplicities over the blow-up of a regular local ring at its closed point in the mixed characteristics. Specifically, it is shown that for an \(n\)-dimensional regular local ring \((R,m)\) with residue field \(K\), then \(H^0(Y,T_{{\mathbb P}^{n-1}}(-1)\otimes {\mathcal O}_Y)\) is \(n\)-dimensional if \(Y\) is the reduced scheme associated to a regular alteration of the special fiber of the blow-up of a subvariety of \(\text{Spec} R\) at the point \([m]\). sheaf cohomology; connectedness; tangent bundles; normal crossing Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Singularities in algebraic geometry, Multiplicity theory and related topics On efficient generation of pull-back of \(T_{\mathbb P^n}(-1)\)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(V\) be a smooth projective variety over a \(p\)-adic field \(k\). Let \(CH_0(V)\) and \(A_0(V)\) be the group of zero-cycles on \(V\) modulo rational equivalence and its subgroup consisting of cycles of degree zero. It is conjectured that \(A_0(V)\) is the direct sum of a finite group and a \(p'\)-divisible group. (An abelian group is called \(p'\)-divisible iff it is divisible by any integer prime to \(p\).) In the paper under review, this conjecture is proved under a weak assumption, namely that \(V\) has a regular projective flat model over the ring of integers in \(k\) on which the reduced subscheme of the special fiber has simple normal crossings. This result has not been known even when \(V\) has good reduction.
This result is deduced from the following strong theorem on a cycle map, which is proved in wider context. Let \(R\) be an excellent henselian discrete valuation ring with residue field \(F\) and fraction field \(k\). (No assumption is made for the characteristic of \(k\).) Let \(X\) be a regular projective flat scheme over \(R\) on which the reduced subscheme of the special fiber has simple normal crossings. We write \(CH_1(X)\) for the Chow group of one-dimensional cycles on \(X\). Let \(n\) be a natural number prime to the characteristic of \(F\). There is a cycle class map \(\rho_X : CH_1(X)/n \to H^{2d}_{et}(X, \mathbb{Z}/n\mathbb{Z}(d))\), where \(d = \dim X - 1\). The following strong theorem is proved in this paper: If \(F\) is finite or separably closed, then \(\rho_X\) is bijective.
This theorem is studied in the framework of Kato homology. We keep the same notation and assumption as the preceding paragraph. The Kato homology \(KH_a(X, \mathbb{Z}/n\mathbb{Z})\) is defined to be the homology group of a complex of the form
\[
\bigoplus_{x \in X_0} H^0_{et}(x, \mathbb{Z}/n\mathbb{Z}(-1)) \leftarrow \cdots \leftarrow \bigoplus_{x \in X_a} H^a_{et}(x, \mathbb{Z}/n\mathbb{Z}(a-1)) \leftarrow \cdots
\]
in degree \(a\). (The leftmost term is placed in degree \(0\).) Let \(l\) be a prime number different from the characteristic of \(F\), and put \(KH_a(X, \mathbb{Q}_l/\mathbb{Z}_l) = \lim_{\to} KH_a(X, \mathbb{Z}/l^n\mathbb{Z})\). The authors propose the following conjectures: If \(F\) is separably closed, then \(KH_a(X, \mathbb{Z}/n\mathbb{Z})\) should vanish for all \(a\). If \(F\) is finite, then \(KH_a(X, \mathbb{Q}_l/\mathbb{Z}_l)\) should vanish unless \(a \not= 1\), and \(KH_1(X, \mathbb{Q}_l/\mathbb{Z}_l)\) should be isomorphic to \((\mathbb{Q}_l/\mathbb{Z}_l)^{\oplus r}\) where \(r\) is the number of the irreducible components of the special fiber of \(X\). (This is a `different weight' analogue of \textit{K. Kato}'s conjecture [J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)].) It is relatively easy to prove the conjectures for \(a \leq 1\). Roughly speaking, the conjecture for \(a=2\) is equivalent to the surjectivity of \(\rho_X\), and when \(\dim X = 3\) the conjecture for \(a=3\) implies the injectivity of \(\rho_X\). As a consequence of the theorem in the preceding paragraph, the conjectures are proved in degree \(a \leq 3\).
The proof of the above results are carried out by induction on the dimension of \(X\). Among main ingredients to proceed the induction step are theorems of Bertini type and of affine Lefschetz type. The first one, due to Uwe Jannsen and the first author, affirms the existence of a `very nice' hypersurface section over \(R\). (This is a refinement of their previous result [\textit{U. Jannsen} and \textit{S. Saito}, ``Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory'', \url{arXiv:0911.1470}, to appear in J. Algebraic Geom.].) The second one asserts that, for the complement of such a `very nice' hypersurface, a vanishing theorem of the etale cohomology holds in a stronger form than Artin-Gabber's affine Lefschetz theorem. Chow group; cycle map; Kato homology; affine Lefschetz; Bertini theorem Saito, Shuji; Sato, Kanetomo, A finiteness theorem for zero-cycles over \textit{p}-adic fields, Ann. of Math. (2), 172, 3, 1593-1639, (2010), With an appendix by Uwe Jannsen, MR 2726095 (2011m:14010) (Equivariant) Chow groups and rings; motives, Local ground fields in algebraic geometry A finiteness theorem for zero-cycles over \(p\)-adic fields
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The first part of the paper focuses on finding a local detection property for projectivity of modules for Frobenius kernels over an algebraically closed field \(k\) of prime characteristic \(p\). For an example of this, consider a finite group \(G\). \textit{L.~Chouinard} [J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)] showed that a \(kG\)-module \(M\) (even infinite dimensional) is projective if and only if it is projective upon restriction to \(kE\) for all elementary Abelian \(p\)-subgroups \(E\) of \(G\). For a `finite' dimensional module, Chouinard's Theorem was later seen to be a consequence of the extensive theory of cohomological support varieties for finite groups. However, Chouinard's Theorem is actually needed to deduce key properties in a general theory of support varieties for arbitrary modules.
The author considers Frobenius kernels of a smooth algebraic group \(G\) over \(k\) to which a theory of support varieties exists for finite dimensional modules. For such modules, it follows that projectivity can be detected by subgroup schemes that are Frobenius kernels of the additive group \(\mathbb{G}_a\). The natural question arises as to whether or not this property can be deduced directly and further for arbitrary modules. Building upon work of the reviewer [Proc. Am. Math. Soc. 129, No. 3, 671-676 (2001; Zbl 0990.20028)] for unipotent algebraic group schemes, the author shows that the detection property holds in general. The author then proceeds to develop a theory of `support cones' for arbitrary modules and uses the above detection result to derive desired properties of these cones. Lastly, the author gives a description of support cones in terms of Rickard idempotent modules as done for finite groups by \textit{D. Benson, J. Carlson}, and \textit{J. Rickard} [Math. Proc. Camb. Philos. Soc. 120, No. 4, 597-615 (1996; Zbl 0888.20003)]. algebraic groups; Frobenius kernels; projectivity; cohomological support varieties; support cones; Rickard idempotent modules; group schemes Pevtsova, Julia, Infinite dimensional modules for Frobenius kernels, J. Pure Appl. Algebra, 0022-4049, 173, 1, 59\textendash86 pp., (2002) Representation theory for linear algebraic groups, Group schemes, Modular Lie (super)algebras, Modular representations and characters, Group rings of infinite groups and their modules (group-theoretic aspects), Group rings Infinite dimensional modules for Frobenius kernels
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The main result of the paper under discussion (Theorem 5.5) is the following variant of the Milnor conjecture for local rings:
if \(A\) is a local ring in which \(1+1\) is invertible, then there is a well-defined group homomorphism
\[e_n : I^n(A) \rightarrow H_{\text{et}}^n(A,\mathbb{Z}/2), \ \text{given by}\]
\[\langle \! \langle a_1,\dots,a_n \rangle \! \rangle \mapsto (a_1) \cup \dots \cup (a_n),\]
whose kernel is exactly \(I^{n+1}(A)\).
A key tool is the proof of the Gersten conjecture for Witt groups (Theorem 3.8) for such \(A\) under the additional assumption that \(A\) be regular and unramified.
In the last section (no. 6), the author provides sufficient conditions for the triviality of the intersection \(\bigcap I^n(A)\) (which in the case of fields follows the Arason-Pfister Hauptsatz), such as \(A\) being a regular ring for which the Gersten conjecture holds. cohomological invariants; quadratic forms; local fields; Etale cohomology; Witt groups Quadratic forms over local rings and fields, Étale and other Grothendieck topologies and (co)homologies Cohomological invariants for quadratic forms over local rings
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author's summary: ``Let \(k\) be a field of characteristic zero. Let \(f: X\to S\) be a proper smooth morphism of non-singular algebraic \(k\)- varieties. If \(k\subset \mathbb{C}\), let \(f^{\text{an}}: X^{\text{an}}\to S^{\text{an}}\) be the analytic morphism associated to \(f\); then the cohomology groups of the fibers, \(H^ i(X^{\text{an}}_ s,\mathbb{C})\), \(s\in S\), form a complex local system over \(S^{\text{an}}\), which is the system of local solutions of the Gauss-Manin connection. By results of Fuchs, Grothendieck, Griffiths,\dots, it is known that this connection has relevant properties: it is defined over \(k\), has regular singular points, its exponents are rational and it supports a variation of Hodge structures. The rational homotopy of the fibers of \(f\) also gives other important complex local systems over \(S^{\text{an}}\). The aim of this article consists in proving that the holomorphic connections associated to every local system that comes from rational homotopy of the fibers have the same algebraic properties than the connection coming from the cohomology''. analytic morphism; Gauss-Manin connection; rational homotopy; cohomology V. NAVARRO AZNAR, Sur la connection de Gauss-Manin en homotopie rationelle, Ann. scient. EÂc. Norm. Sup. 26 (1993), 99-148. Rational homotopy theory, de Rham cohomology and algebraic geometry Sur la connexion de Gauss-Manin en homotopie rationnelle. (On the Gauss- Manin connection in rational homotopy)
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\phi\) be a rational map \(\phi: \mathbb P^n\cdots\to S \subset \mathbb P^{n+1}\), where \(S\) is a normal hypersurface and \(\phi\) is birational onto the image. \(\phi\) has bidegree \((d',d)\) if it is given by a subsystem of \(|\mathcal O_{\mathbb P^n}(d')|\) and its inverse is given by a subsystem of \(|\mathcal O_S(d)|\). The author studies maps \(\phi\) of bidegree \((2,d)\), for which the base locus \(B\) is smooth and connected.
When \(d=1\), there is only one example of such rational map \(\phi\) (modulo projective equivalence). When \(d>1\), the author studies the classification of maps \(\phi\), in terms of the degree of the hypersurface \(S\). In particular, the author describes the structure of maps \(\phi\) of bidegree \((2,2)\), when \(\deg(S)=2\). Weaker results when both \(d\) and \(\deg(S)\) grow are listed. The author also shows how one can find a classification of maps of bidegree \((2,d)\), when some restrictions on the base locus \(B\) (e.g. restrictions on the dimension of \(B\)) are introduced. quadratic transformations Staglianò, G.: On special quadratic birational transformations of a projective space into a hypersurface. Rend. circ. Mat. Palermo 61, No. 3, 403-429 (2012) Rational and birational maps On special quadratic birational transformations of a projective space into a hypersurface
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let (A,\({\mathfrak m})\) be a local ring of Krull dimension 2. We call R and G respectively the extended Rees algebra and the form ring of A with respect to \({\mathfrak m}\). G is assumed to be a normal integral domain; then Cl(A)\(\cong Cl(R)\). Moreover, following \textit{J. Lipman} [Am. J. Math. 101, 203-211 (1979; Zbl 0417.13009)] there exists a canonical morphism \(j: Cl(R)\to Cl(G).\) The paper contains (among other results which give perspective) the proofs of the following facts: \((i)\quad if\) A is henselian, then j is surjective; \((ii)\quad if\quad (H^ 2_{G_ +}(G))_ n=0\) for every \(n>0\), then j is injective.
The proofs use only methods of commutative algebra and local cohomology theory. divisor class groups of a two-dimensional local ring; extended Rees algebra; form ring; local cohomology theory Ideals and multiplicative ideal theory in commutative rings, Local cohomology and algebraic geometry, Local rings and semilocal rings, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Dimension theory, depth, related commutative rings (catenary, etc.) On the divisor class groups of a two-dimensional local ring and its form ring
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\subset\mathbb{P}^r\) be a smooth linearly normal variety of dimension \(n\geq 2\) such that \(h^1(\mathcal O_X)=0\). In this paper the authors prove that, if the degree of \(X\) is \(d\leq 2(r-n)-1\), the quadrics through \(X\) give a special subhomaloidal system (a special homaloidal system if \(\text{Sec}(X)\subset\mathbb{P}^r\)). Moreover, if \(d\leq 2(r-n)\), such an \(X\) is the scheme-theoretic intersection of the quadrics containing it.
The proof is based on Castelnuovo's argument that a set of \(2s+1\) points in general linear position imposes independent conditions on quadric hypersurfaces.
These results are applied to the study of some varieties with one apparent double point. Projective techniques in algebraic geometry, Special surfaces The usual Castelnuovo's argument and special subhomaloidal systems of quadrics
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author generalizes his paper [Rev. Mat. Iberoam. 25, No. 3, 995--1054 (2009; Zbl 1207.14009)] which gave an algorithmic resolution of singularities in characteristic zero. Here he develops a theory of simultaneous resolution completing his precedent results. The algorithm is an adaptation of the algorithm of \textit{S. Encinas} and \textit{O. Villamayor} [Prog. Math. 181, 147--227 (2000; Zbl 0969.14007)] and involves multi-ideals wich are essentially a pair consisting of a sheaf of ideals and a positive integer.
This algorithm is classical if the object are defined over a field but has a real interest if they are defined over an artinian ring. resolution of singularities; multi-ideals; equiresolution; artinian rings Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Computational aspects of higher-dimensional varieties, Commutative Artinian rings and modules, finite-dimensional algebras Resolution algorithms and deformations
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In a previous article [Arch. Math. 66, No. 2, 163-176 (1996; see the preceding review)], the second and fifth authors described a generalization of the algorithms of Buchberger and Mora to the case of any ordering on the monomials in a polynomial ring that is compatible with the semigroup structure. (The variables may have negative, positive, or zero weights.) In the present article, the authors describe the implementation of these ideas in the system SINGULAR.
In the first part of the paper, the implementation of the standard basis algorithm is presented. Two methods that can improve the efficiency of this algorithm are the HCtest and ecartMethod. The HCtest is applicable only to 0-dimensional ideals and involves computing the minimal monomial not contained in the initial ideal (the highest corner) and discarding all bigger monomials in further computations. The ecartMethod is applicable in all cases and provides an automatic choice of a weight vector of positive integers such that the weight ecart of the input polynomials become as small as possible. Results of twenty standard basis computations with various options (e.g. Lazard's method, Mora's algorithm with sugar, HCtest, ecartMethod) are given.
In the last part of the paper, the authors compare different methods for computing syzygies and minimal resolutions in twenty cases. Included in the first implementation of F.-O. Schreyer's algorithm for computing syzygies. No single strategy is best in all cases. system SINGULAR; standard basis algorithm; HCtest; syzygies; minimal resolutions Grassmann, H; Greuel, G-M; Martin, B; Neumann, W; Pfister, G; Pohl, W; Schönemann, H; Siebert, T, Standard bases, syzygies and their implementation in SINGULAR, Appl. Algebra Eng. Commun. Comput., 7, 235-249, (1996) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Symbolic computation and algebraic computation On an implementation of standard bases and syzygies in SINGULAR
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For the entire collection see Zbl 0614.00007.]
The paper studies the local rings A of algebroid curve singularities whose total quotient field is of the form \(A'=k[[ t_ 1]]\times...\times k[[ t_ d]]\) (with k a field). Specifically, one first associates to such an A its semigroup of values \(S=\{(ord_{t_ 1}(pr_ 1(z)),...,ord_{t_ d}(pr_ d(z)))\) with z non-zero divisor in \(A\}\). Then one constructs a monomial subring \(A_ S\) of \(k[[ t_ 1]]\times...\times k[[ t_ d]]\) which is saturated and has S as semigroup of values. At least in characteristic zero \(A_ S\) is precisely the saturation of A. algebroid curve singularities; semigroup of values; saturation Delgado, F.; Núñez, A.: Monomial rings and saturated rings, Travaux en cours 22 (1987) Singularities of curves, local rings, Local ground fields in algebraic geometry, Formal power series rings, Singularities in algebraic geometry, Arithmetic ground fields for curves Monomial rings and saturated rings
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system \textit{I. V. Dolgachev} and \textit{B. Yu. Weisfeiler} [Izv. Akad. Nauk SSSR Ser. Mat. 38, 757--799 (1974; Zbl 0314.14015)] formulated the following
Conjecture. Let \(f:X\to S\) be a flat affine morphism of smooth schemes with every fiber isomorphic (over the residue field) to an affine space. Then \(f\) is locally trivial in the Zariski topology.
In the characteristic-0 case, this conjecture is known to be true (under much weaker assumptions) for morphisms of relative dimension 1. The known partial positive results in higher relative dimensions (see e.g., \textit{A. Sathaye} [Invent. Math. 74, 159--168 (1983; Zbl 0538.13006)] and \textit{T. Asanuma} and \textit{S. M. Bhatwadekar} [J. Pure Appl. Algebra 115, 1--13 (1997; Zbl 0893.13006)]) deal only with families over a 1-dimensional base with 2-dimensional fibers, under an extra assumption that the generic fiber is the affine plane as well. In this paper we show that the latter assumption holds over any base. To simplify consideration, we restrict it to smooth, quasi-projective varieties defined over \(\mathbb{C}\). We say that a family \(f:X\to S\) of quasi-projective varieties contains a cylinder if, for some Zariski open subset \(S_0\) of \(S\), there is a commutative diagram
\[
\begin{matrix} f^{-1}(S_0) & \overset\varphi \longrightarrow & S_0\times\mathbb{C}^k\\ f \searrow && \swarrow \text{pr}_1\\ & S_0\end{matrix}
\]
where \(\varphi\) is an isomorphism. Our main result is the following
Theorem 1. A smooth family \(f:X\to S\) with general fibers isomorphic to \(\mathbb{C}^2\) contains a cylinder \(S_0\times \mathbb{C}^2\).
We do not know if the theorem remains true in higher relative dimensions. A theorem of Sathaye [loc. cit.], together with theorem 1, proves the following.
Corollary. The Dolgachev-Weisfeiler conjecture is indeed true for families of affine planes over smooth curves.
On the hand, theorem 1 provides one of the principal ingredients in the proof of the following statement.
Theorem [\textit{S. Kaliman}, Pac. J. Math. 203, No. 1, 161--190 (2002; Zbl 1060.14085)]. A polynomial \(p\) on \(\mathbb{C}^3\) with general fibers isomorphic to \(\mathbb{C}^2\) is a variable of the polynomial algebra \(\mathbb{C}^{[3]}\) (that is, \(\mathbb{C}^{[3]}\simeq \mathbb{C}[p]^{[2]})\). In particular, all its fibers are isomorphic to \(\mathbb{C}^2\). B. _. Ve_sfe_ler, I. V. Dolgaqev, Unipotentnye shemy grupp nad celostnymi kol\ ~\ cami, Izv. AN SSSR, Ser. mat. 38 (1974), vyp. 4, 757-799. Engl. transl.: B. J. Veĭsfeĭler, I. V. Dolgačev, Unipotent group chemes over integral rings, Math. USSR-Izv. 8 (1974), no. 4, 761-800. Affine fibrations, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Families of affine planes: the existence of a cylinder.
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let G be a reductive algebraic group acting (linearizably) on projective varieties X and Y, and let \(\pi:\quad Y\to X\) be a G-equivariant morphism. The author defines a suitable linearization of the G-action on Y and then compares stability in X and Y. His most general result is that \(q\in Y\) is unstable (resp. properly stable) if \(\pi\) (q) is unstable (resp. properly stable). The most important result of the paper is a relative Hilbert-Mumford theorem: it states that stability of \(q\in Y\) and \(\pi\) (q)\(\in X\) can be tested simultaneously by the weights of 1- parameter subgroups of G.
If \(\pi\) : \(Y\to X\) is a blowing up the author completely describes the unstable and properly stable loci in Y in terms of \(\pi\) (outside the exceptional divisor in general, and under certain smoothness hypothesis even everywhere). The last theorems of the paper deal with the ``strictly semi-stable'' locus \(X^{sss}\) in X, i.e. the semi-stable but not properly stable points: it is shown that \(X^{sss}\) is singular or empty if the semi-stable locus \(X^{ss}\) is smooth and that a resolution of singularities of \(X^{sss}\) yields a ``stable resolution'' of X, i.e. a G-variety \(\tilde X\) where each point is either unstable or properly stable. linear action of reductive algebraic group; stable points; 1-parameter subgroups; \(X^{sss}\); resolution of singularities Zinovy Reichstein, Stability and equivariant maps, Invent. Math. 96 (1989), no. 2, 349-383. Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects) Stability and equivariant maps
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\) be a paracompact locally-ringed space over a field \(C\) of sufficiently large or zero characteristic. Let \(E\) be a simple locally free sheaf over \(X\). The author gives a canonical construction for the universal \(m\)-th order deformation space of \(E\) and for the Poincaré bundle over it. The construction involves the Jacobi complexes \(J^\bullet_m(sl (E))\) whose definition is briefly recalled. The above deformation space turns out to be \(\text{Spec}(C\oplus H^0(J^\bullet_m(sl (E)))\).
The proof uses the Kodaira-Spencer bi-complexes associated to a deformation of \(E\) and it is only sketched.
The author claims that the same construction can be applied to deformation of manifolds, yielding a simplification of the author's preprint ``Canonical infinitesimal deformations''.
An interesting application, stated in full generality, is the closedness of the trace forms, that are \(H^2({\mathcal O}_X)\)-valued \(2\)-forms on a germ parametrizing a deformation of a simple locally free sheaf over \(X\).
A striking consequence of this closedness theorem is a unified proof of the following two known results:
(i) the existence of a symplectic structure on the moduli space of bundles over a compact complex surface \(S\) with \(K_S={\mathcal O}_S\) [\textit{S. Mukai}, Invent. Math. 77, 101-116 (1984; Zbl 0565.14002)],
(ii) the existence of a symplectic structure on the moduli space of local systems over a compact Riemann surface [\textit{N. J. Hitchin}, Common trends in mathematics and quantum field theories, Kyoto and Tokyo 1990, Prog. Theor. Phys., Suppl. 102, 159-174 (1990; Zbl 0793.53033)]. deformation space; Poincaré bundle; deformation of a simple locally free sheaf; symplectic structure on the moduli space bundles; compact complex surface; Riemann surface Z. RAN, On the local geometry of moduli spaces of locally free sheaves, Moduli of vector bundles (Sanda, 1994; Kyoto, 1994), pp. 213-219, Lecture Notes in Pure and Appl. Math., 179 (Dekker, New York, 1996). Zbl0884.14005 MR1397990 Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Formal methods and deformations in algebraic geometry, Families, moduli, classification: algebraic theory, Riemann surfaces; Weierstrass points; gap sequences, Compact complex surfaces, General geometric structures on manifolds (almost complex, almost product structures, etc.) On the local geometry of moduli spaces of locally free sheaves
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A common way of studying algebraic varieties is to study their embeddings in projective space by linear series of base-point free line bundles. In this article, this technique is extended to multigraded linear series of a collection of globally generated vector bundles on a scheme, unifying several constructions in the study of algebraic varieties. The primary goal is to describe schemes as a moduli space to achieve geometric properties. The article gives several explicit families of examples.
Throughout the article, \(\Bbbk\) denotes an algebraically closed field of characteristic \(0\). Let \(Y\) be a projective scheme. Given a collection \(E_1,\dots,E_n\) of effective vector bundles on \(Y\), let \(E=\bigoplus_{0\leq i\leq n}E_i\) where \(E_0\simeq\mathcal O_Y\). Let \(A=\text{End}_Y(E)\) be the endomorphism algebra of \(E\) and let \(\mathbf{v}=(v_i)\) with \(v_i=\text{rk}(E_i)\) be the dimension vector. The \textit{multigraded linear series} of \(E\) is the fine moduli space \(\mathcal M(E)\) of \(0\)-generated \(A\)-modules with dimension vector \(\mathbf{v}\). The universal family of \(\mathcal M(E)\) is a vector bundle \(T=\bigoplus_{0\leq i\leq n}T_i\) together with a \(\Bbbk\)-algebra homomorphism \(A\rightarrow\text{End}(T)\) where each \(T_i\) is a tautological vector bundle of rank \(v_i\) and \(T_0\) is the trivial line bundle.
The first main result of the article generalizes the classical morphism \(\phi_{|L|}:Y\rightarrow |L|\) to the linear series of a single base-point free line bundle \(L\) on \(Y\), i.e. the morphism to a Grassmannian defined by a globally generated vector bundle on a projective variety. Verbatim:
Theorem 1.1. If the vector bundles \(E_1,\dots,E_n\) are globally generated, then there is a morphism \(f:Y\rightarrow\mathcal M(E)\) satisfying \(E_i=f^\ast(T_i)\) for \(0\leq i\leq n\) whose image is isomorphic to the image of the morphism \(\phi_{|L|}:Y\rightarrow|L|\) to the linear series of \(L:=\bigotimes_{1\leq i\leq n}\det(E_i)^{\otimes j}\) for some \(j>0.\)
If the line bundle \(\bigotimes_{1\leq i\leq n}\det(E_i)\) is ample, after taking a multiple of a linearisation if necessary, the resulting universal morphism \(f:Y\rightarrow\mathcal M(E)\) is a closed immersion. The next question is then if \(f\) is surjective. If that is the case, then \(f\) represents \(Y\) as the fine moduli space \(\mathcal M(E)\). It turns out that even when \(Y\) is isomorphic to \(\mathcal M(E),\) more insight can be gained by deleting some summands of \(E.\) When \(0\in C\subseteq\{0,1,\dots,n\},\) the subbundle \(E_C=\sum_{i\in C}E_i\) has the trivial subbundle \(E_0\) as a summand, and Theorem 1.1 proves the universal morphism \(g_C:\mathcal M(E)\rightarrow\mathcal M(E_C)\) between multigraded linear series. This can lead to a more geometric significant moduli space description of \(Y\).
Of such geometric significance is a moduli construction determined by a tilting bundle. \textit{A. Bergman} and \textit{N. J. Proudfoot} [Pac. J. Math. 237, No. 2, 201--221 (2008; Zbl 1151.18011)] proved that for a smooth variety \(Y\) with a tilting bundle \(E\), \(f\) is an isomorphism onto a connected component of \(\mathcal M(E)\). A main goal of this article is to establish several cases where \(f\) is an isomorphism onto \(\mathcal M(E)\) itself, implying a description of \(Y\) as a moduli space. Also, the authors manage to give results in situations where \(Y\) is singular.
A second main goal is to apply the theory to the \textit{special McKay correspondence}. Let \(G\subset\text{GL}(2,\Bbbk)\) be a finite subgroup without pseudo-reflections, let \(\text{Irr}(G)\) be the set of isomorphism classes of irreducible representations of \(G\), and let \(Y\) denote the minimal resolution of \(\mathbb A^2_{\Bbbk}/G\). \textit{R. Kidoh} [Hokkaido Math. J. 30, No. 1, 91--103 (2001; Zbl 1015.14004)] and \textit{A. Ishii} [J. Reine Angew. Math. 549, 221--233 (2002; Zbl 1057.14057)] proved that \(Y\) is isomorphic to the fine moduli space of \(G\)-equivariant coherent sheaves of the form \(\mathcal O_Z\), for subschemes \(Z\subset\mathbb A^2_{\Bbbk}\) such that \(\Gamma(\mathcal O_Z)\) is isomorphic to the regular representation of \(G\) (\(G\)-Hilbert scheme). Writing the tautological bundle on the \(G\)-Hilbert space \(T=\underset{\rho\in\text{Irr}(G)}{\sum} T_\rho^{\bigoplus\dim(\rho)}\) and noticing that \(\text{End}_Y(T)\) is isomorphic to the skew group algebra, it follows that the minimal resolution \(Y\cong G-\text{Hilb}\) is isomorphic to the multigraded linear series \(\mathcal M(T)\). When \(G\) is a finite subgroup of \(\text{SL}(2,\Bbbk)\), \(T\) is a tilting bundle on \(Y\) so that \(Y\) is derived equivalent to the category of modules over \(\text{End}_Y(T),\) but the authors prove that this is false in general.
A natural moduli description of \(Y\) is given by the special McKay correspondence of the finite subgroup \(G\subset\text{GL}(2,\Bbbk)\). For the set \(\text{Sp}(G)=\{\rho\in\text{Irr}(G)|H^1(T^\vee_\rho)=0\}\) of special representations, Van den Bergh proved that the \textit{reconstruction bundle} \(E:=\underset{\rho\in\text{Sp}(G)}{\bigoplus} T_\rho\) is a tilting bundle on \(Y\) so that \(Y\) is derived equivalent to the category of modules over the reconstruction algebra studied by \textit{M. Wemyss} [Math. Ann. 350, No. 3, 631--659 (2011; Zbl 1233.14012)]. The second main result of the article proves that \(E\) contains enough information to reconstruct \(Y\), and provides a moduli space description that trumps the \(G\)-Hilbert scheme in general. Verbatim:
Theorem 1.2. Let \(G\subset\text{GL}(2,\Bbbk)\) be a finite subgroup without pseudo-reflections. Then: (i) the minimal resolution \(Y\) of \(\mathbb A^2_{\Bbbk}/G\) is isomorphic to the multigraded linear series \(\mathcal M(E)\) of the reconstruction bundle; and (ii) for any partial resolution \(Y^\prime\) such that the minimal resolution \(Y\rightarrow\mathbb A^2_\Bbbk/G\) factors via \(Y^\prime,\) there is a summand \(E_C\subseteq E\) such that \(Y^\prime\) is isomorphic to \(\mathcal M(E_C).\)
The authors correctly remark that the approach in this article is closer in spirit to the geometric construction of the special McKay correspondence for cyclic subgroups of \(\text{GL}(2,\Bbbk)\) given by \textit{A. Craw} [Q. J. Math. 62, No. 3, 573--591 (2011; Zbl 1231.14010)] in his earlier work.
The main tools for proving the theorems is a homological criterion to decide when the morphism \(g_C\) is surjective. Any subset \(C\subseteq\{0,\dots,n\}\) containing \(0\) determines a subbundle \(E_C\) of \(E\), and the module categories of the algebras \(A=\text{End}_Y(E)\) and \(A_C=\text{End}_Y(E_C)\) are linked by \textit{recollement}. The authors prove that the morphism \(g_C:\mathcal M(E)\rightarrow\mathcal M(E_C)\) is surjective iff for each \(c\in\mathcal M(E_C)\), the \(A\)-module \(j_!(N_x)\) admits a surjective map onto an \(A\)-module of dimension vector \(\mathbf(v)\).
As a second main ingredient, the authors use the fact that a derived equivalence \(\Psi(-)=E^\vee\otimes_A-:D^b(A)\rightarrow D^b(Y)\) induces an isomorphism between the lattice of dimension vectors for \(A\) and the numerical Grothendieck group for compact support \(K_c^{\text{num}}(Y)\) introduced by Bayer-Craw-Zhang.
The article concludes with examples from NCCRs in dimension three. In very short terms, a resolution \(Y\rightarrow\mathbb A/G\) is given. Examples are given where one can reconstruct \(Y\) using only a proper summand of a tilting bundle \(T\).
The article is very important, contains very nice results with clear proofs, and show important applications of tilting- and moduli theory, and their connection. linear series; base-point free line bundles; effective vector bundles; multigraded linear series; summands; special McKay correspondance; pseudo-reflections; irreducible group-representations; G-Hilbert space; skew group algebra; derived equivalence; reconstruction bundle; reconstruction algebra; numerical Grothendieck group; NCCRs Craw, A., Ito, Y., Karmazyn, J.: Multigraded linear series and recollement (2017). arXiv:1701.01679 (to appear in Math. Z.) Noncommutative algebraic geometry, McKay correspondence, Representations of quivers and partially ordered sets, Grothendieck groups (category-theoretic aspects) Multigraded linear series and recollement
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Consider a normal variety \(X\) with only constant invertible global functions and finitely generated free divisor class group \(\text{Cl}(X)\). Then one can associate a natural object to it, that is, the Cox ring. Namely, choose a subgroup \(K\) of the group of Weil divisors mapping isomorphically to \(\text{Cl}(X)\). Then the Cox ring is a ring
\[
\bigoplus_{D\in K} \Gamma (X, \mathcal O(D)).
\]
The authors study the properties of the variety in terms of the combinatorics of its Cox ring.
From the abstract:
Given a variety \(X\) with a finitely generated total coordinate ring, we describe basic geometric properties of \(X\) in terms of certain combinatorial structures living in the divisor class group of \(X\). For example, we describe the singularities, we calculate the ample cone, and we give simple Fano criteria. As we show by means of several examples, the results allow explicit computations. As immediate applications we obtain an effective version of the Kleiman-Chevalley quasiprojectivity criterion, and the following observation on surfaces: a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non-factorial singularities admit a common affine neighbourhood. Cox ring; bunched ring; Fano variety; \(A_2\)-maximality Berchtold, Florian; Hausen, Jürgen, Cox rings and combinatorics, Trans. Amer. Math. Soc., 0002-9947, 359, 3, 1205-1252 (electronic), (2007) Divisors, linear systems, invertible sheaves, Fano varieties, Hypersurfaces and algebraic geometry, Rational and unirational varieties, Toric varieties, Newton polyhedra, Okounkov bodies, Computational aspects of higher-dimensional varieties Cox rings and combinatorics
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors examine the Poisson geometry of moduli spaces of local systems.
Let \(X\) be a smooth complex variety of dimension \(d\) and \(G\) a reductive group. If \(X\) is curve than it is well-known that the moduli space of \(G\)-local systems on \(X\) carries a canonical Poisson structure, whose symplectic leaves are moduli of \(G\)-local systems whose monodromy at infinity is fixed (up to conjugacy).
The authors give a natural extension of this result to the case of higher dimensions, however even the correct formulation now invariably involves derived geometry. If \(d > 1\) then the moduli space of \(G\)-local systems is naturally replaced by a derived moduli stack \(\mathrm{Loc}_G(X)\) of \(G\)-local systems (note that this takes into account the whole homotopy type of \(X\) as opposed to just the fundamental group). It follows from earlier results of the authors and their collaborators that for any compact oriented (real) manifold \(M\) there is a \((2 - \dim_{ \mathbb R} M)\)-shifted symplectic structure on \(\mathrm{Loc}_G(M)\) and thus a non-degenerate \((2 - \dim_{ \mathbb R} M)\)-shifted Poisson structure.
In this new paper the authors show that if \(X\) is a smooth complex variety, not necessarily proper, \(\mathrm{Loc}_G(X)\) carries a canonical \((2-2d)\)-shifted Poission structure.
They moreover describe some generalized symplectic leaves of the foliation if \(X\) admits a smooth compactification whose divisor at infinity is simple normal crossing with at most double intersections. Analogously to the case of curves these leaves are given by derived moduli of \(G\)-local system with fixed local monodromy at infinity as long as a technical condition the authors call `strictness' is satisfied.
As the authors say this is a first step towards understanding moduli of local systems on higher dimensional open varieties, with nonproper generalizations of Simpson's nonabelian Hodge theory as a key long term motivation.
The key ingredient in the proof is the restriction map to the boundary at infinity. Any smooth complex algebraic variety has well-defined boundary at infinity \(\partial X\) that is a compact manifold of real dimension \(2d-1\). Now \(\mathrm{Loc}_G(\partial G)\) has a shifted symplectic structure and by results of Calaque the restriction map is Lagrangian [\textit{D. Calaque}, Contemp. Math. 643, 1--23 (2015; Zbl 1349.14005)]. Then this induces the Poisson structure on \(\mathrm{Loc}_G(X)\) by \textit{V. Melani} and \textit{P. Safronov} [Sel. Math., New Ser. 24, No. 4, 3061--3118 (2018; Zbl 1461.14006); Sel. Math., New Ser. 24, No. 4, 3119--3173 (2018; Zbl 1440.14004)]. The characterization of symplectic leaves is more involved, already the notion of `fixing the monodromy at infinity' is more complicated than in the 1-dimensional case.
The paper includes useful discussions of algebraic descriptions of the moduli of \(G\)-local systems and of the boundary at infinity of a smooth complex variety. The special case that \(X\) is a curve is discussed in detail. The authors expect (but do not show) that in this case the 0-shifted symplectic structure they construct agrees with the one that is known from the literature. local systems; shifted symplectic structures; derived moduli stacks Families, moduli, classification: algebraic theory, Stacks and moduli problems, Symplectic structures of moduli spaces Poisson geometry of the moduli of local systems on smooth varieties
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\mathbf f=(f_{1}, \dots , f_{l}):U\rightarrow K^{l}\), with \(K=\mathbb R\) or \(\mathbb C\), be a \(K\)-analytic mapping defined on an open set \(U\subset K^{n}\), and let \(\Phi \) be a smooth function on \(U\) with compact support. In this paper, we give a description of the possible poles of the local zeta function attached to \((\mathbf f, \Phi )\) in terms of a log-principalization of the ideal \(\mathcal I_{f}=(f_{1}, \dots , f_{l})\). When \(\mathbf f\) is a non-degenerate mapping, we give an explicit list for the possible poles of \(Z_{\Phi}(s, \mathbf f)\) in terms of the normal vectors to the supporting hyperplanes of a Newton polyhedron attached to f, and some additional vectors (or rays) that appear in the construction of a simplicial conical subdivision of the first orthant. These results extend the corresponding results of \textit{A. N. Varchenko} [Funct. Anal. Appl. 10, 175-196 (1977); translation from Funkts. Anal. Prilozh. 10, No. 3, 13--38 (1976; Zbl 0351.32011)] to the case \(l\geqslant 1\), and \(K=\mathbb R\) or \(\mathbb C\). In the case \(l=1\) and \(K=\mathbb R\), \textit{J. Denef} and {P. Sargos} [J. Anal. Math. 53, 201--218 (1989; Zbl 0693.32003)] proved that the candidate poles induced by the extra rays required in the construction of a simplicial conical subdivision can be discarded from the list of candidate poles. We extend the Denef-Sargos result to arbitrary \(l\geqslant 1\). This yields, in general, a much shorter list of candidate poles, which can, moreover, be read off immediately from \(\Gamma (\mathbf f)\). local zeta functions; poles Edwin León-Cardenal, Willem Veys & Wilson A. Zúñiga-Galindo, Poles of Archimedean zeta functions for analytic mappings, J. Lond. Math. Soc.87 (2013), p. 1-21 Local complex singularities, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Poles of Archimedean zeta functions for analytic mappings
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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Given an \(n\)-dimensional isolated complete intersection singularity (icis) \((X,0)= (F^{-1}(0),0)\subset (\mathbb{C}^{n+ k},0)\) where \(F= (f_1,\dots, f_k): (\mathbb{C}^{n+k},0)\to (\mathbb{C}^k,0)\), the authors study its \(p\)-fold suspension \((X^{(p)},0)\subset (\mathbb{C}^{n+k+1},0)\) defined by \((f_1+ a_1z^p,\dots, f_k+ a_kz^p)\), where \(a_1,\dots, a_k\) are generic coefficients.
In the case of the \(p\)-fold suspension \((X^{(p)},0)\) of an icis \((X,0)\subset (\mathbb{C}^2,0)\) defined by two generic function germs \(f,g: (\mathbb{C}^2,0)\to (\mathbb{C},0)\), they compute the following data:
1) a distinguished system of generators of the vanishing homology group \(H_n(V_\varepsilon,\mathbb{Z})\) (the ``Milnor lattice''), where \(V_\varepsilon\) is the Milnor fibre,
2) the equivariant intersection numbers between these cycles, and thus the Coxeter-Dynkin diagram of \((X^{(p)},0)\), and
3) the \(\mathbb{Z}\)-linear relations between these cycles.
On the way, they also prove an equivariant analogue of the Picard-Lefschetz formula for the \(A_{p-1}\)-singularity. The paper ends with some examples of Coxeter-Dynkin diagrams of suspensions over the intersections of germs of some particular plane curves. isolated complete intersection singularity; suspension; Coxeter-Dynkin diagrams Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Germs of analytic sets, local parametrization, Singularities in algebraic geometry, Complete intersections Suspensions of fat points and their intersection forms
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