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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 goal of this paper is the following theorem: A normal two-dimensional singularity (V,p) of multiplicity two over \({\mathbb{C}}\) satisfies the condition \(p_ a(V,p)\leq 1\) if and only if each normalization in Zariski's canonical resolution of (V,p) is trivial or obtained by blowing-up along (reduced) \({\mathbb{P}}^ 1\) in the singular locus. This is an analogy of the well-known characterization of the rational double point by the absolute isolatedness. Our result is proved as a corollary of four lemmas; an inequality about \(p_ a\), a formula for computation of the geometric genus due to \textit{E. Horikawa} [Invent. Math. 31, 43-85 (1975; Zbl 0317.14018)], a criterion for the condition \(p_ a\leq 1\) due to \textit{S. S.-T. Yau} [Trans. Am. Math. Soc. 257, 269-329 (1980; Zbl 0465.32008)], and an elementary computation on one-dimensional singularity with multiplicity \(\leq 5\). The effective use of the arithmetic genus \(p_ a\) of the singularity is new at all. arithmetic genus of singularity; normal two-dimensional singularity; resolution Tomari, M., A geometric characterization of normal two-dimensional singularities of multiplicity two with pa^ 1, Publ. R. J. M. S. Kyoto Univ., 20 (1984), 1-20. Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) A geometric characterization of normal two-dimensional singularities of multiplicity two with \(p_ a\leq 1\) | 0 |
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 Author's abstract: Let \(M\) be an analytic manifold over \(\mathbb R\) or \(\mathbb C\), \(\theta\) a 1-dimensional log-canonical (resp. monomial) singular distribution and \(\mathcal I\) a coherent ideal sheaf defined on \(M\). We prove the existence of a resolution of singularities for \(\mathcal I\) that preserves the log-canonicity (resp. monomiality) of the singularities of \(\theta\). Furthermore, we apply this result to provide a resolution of a family of ideal sheaves when the dimension of the parameter space is equal to the dimension of the ambient space minus one. resolution of singularities; singular foliations; log-canonical foliations; monomial foliations Belotto, A, Global resolution of singularities subordinated to a \(1\)-dimensional foliation, J. Algebra, 447, 397-423, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Singularities of holomorphic vector fields and foliations, Foliations in differential topology; geometric theory, Global theory of complex singularities; cohomological properties, Dynamical aspects of holomorphic foliations and vector fields, Dynamics induced by flows and semiflows, Real-analytic and Nash manifolds, Singularities in algebraic geometry Global resolution of singularities subordinated to a 1-dimensional foliation | 0 |
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 \(R=k[x_1,\dots,x_n]\) denote a polynomial ring and let \(h:\mathbb{N} \to\mathbb{N}\) be a numerical function. Consider the set of all graded Artin level quotients \(A=R/I\) having Hilbert function \(\underline h\). This set (if nonempty) is naturally in bijection with the closed points of a quasiprojective scheme \({\mathfrak L}^\circ (\underline h)\). The object of this note is to prove some specific geometric properties of these schemes, especially for \(n=2\). The case of Gorenstein Hilbert functions (i.e., where \(A\) has type 1) has been extensively studied, and several qualitative and quantitative results are known. Our results should be seen as generalizing some of them to the non-Gorenstein case. We derive an expression for the tangent space to a point of \({\mathfrak L}^\circ(\underline h)\). In the case \(n=2\), we give a geometric description of a point of \({\mathfrak L}^\circ (\underline h)\) in terms of secant planes to the rational normal curve, which generalizes the one just given for \(t=1\). We relate this description to Waring's problem for systems of algebraic forms and solve the problem for \(n=2\). In the last section we prove a projective normality theorem for a class of schemes \({\mathfrak L}(i,r)\) using spectral sequence techniques. The results are largely independent of each other, so they may be read separately. DOI: 10.1307/mmj/1049832900 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Classical problems, Schubert calculus On parameter spaces for Artin level algebras. | 0 |
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, \(V\) an \(n\)-dimensional vector space, \(A=\text{Sym}(V)\cong K[x_1,\dots,x_n]\) the polynomial ring in \(n\) variables over the field \(K\) and \(\text{GL}(V)\) the general linear group, which can be viewed as an algebraic group over \(K\). For a partition \(\alpha\), one denotes by \(\mathbf S_\alpha V\) the irreducible representation of \(\text{GL}(V)\) of highest weight \(\alpha\). Let \(\alpha,\beta\) be two partitions such that \(\alpha_1<\beta_1\) and \(\alpha_i=\beta_i\) for all \(i>1\).
\textit{D. Eisenbud, G. Fløystad} and \textit{J. Weyman}, [Ann. Inst. Fourier 61, No. 3, 905-926 (2011; Zbl 1239.13023)], constructed the minimal free resolution of the cokernel of a nonzero map \(\varphi(\alpha,\beta)\colon A\otimes\mathbf S_\beta V\to A\otimes\mathbf S_\alpha V\). The authors give a simpler proof and extend the construction by removing the restrictions on the partitions \(\alpha\) and \(\beta\). Moreover, they give a simple combinatorial algorithm for writing down a (non-minimal, in general) free resolution of the cokernel of a nonzero map of the form \(\varphi(\alpha,\beta^1,\dots,\beta^r)\colon\bigoplus_{i=1}^r A\otimes\mathbf S_{\beta^i}V\to A\otimes \mathbf S_\alpha V\).
The authors consider also the problem of replacing \(A\) by the exterior algebra \(B=\bigwedge V\). They show that, in this case, the resolution is still simple to describe combinatorially, even if it is infinite in length. The results from Eisenbud, Fløystad and Weyman [loc. cit.] are generalized for several classical groups such as orthogonal or symplectic groups.
Finally, the authors consider the equivariant analogue of the Boij-Söderberg decomposition and they show that the equivariant analogue of the Boij-Söderberg algorithm for writing Betti tables as linear combinations of pure Betti tables does not hold. general linear groups; irreducible representations; minimal resolutions; Betti tables; Boij-Söderberg theory; symmetric functions; Schur functors; pure resolutions; equivariant resolutions; Betti diagrams; determinantal varieties Sam, S. V; Weyman, J., \textit{Pieri resolutions for classical groups}, J. Algebra, 329, 222-259, (2011) Representation theory for linear algebraic groups, Syzygies, resolutions, complexes and commutative rings, Determinantal varieties, Classical groups (algebro-geometric aspects) Pieri resolutions for classical groups. | 0 |
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 (\({\mathfrak O},m,k)\) be a complete one-dimensional reduced local ring of equicharacteristic zero. It is known that the Zariski saturation \(\tilde{\mathfrak O}^ Z\) [\textit{O. Zariski}, Am. J. Math. 90, 961-1023 (1968; Zbl 0189.214)] and the Campillo saturation \(\tilde{\mathfrak O}^ C\) [\textit{A. Campillo} in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 211-220 (1983; Zbl 0553.14013)] of \({\mathfrak O}\) agree when k is algebraically closed. Furthermore, in this case, knowledge of either saturation of \({\mathfrak O}\) is equivalent to knowledge of the equisingularity class of a generic plane projection of Spec(\({\mathfrak O})\). The present paper considers the case where k is not algebraically closed. In this case the two saturations differ. Indeed, if \(\bar k\) is the algebraic closure of k and \(k\subseteq {\mathfrak O}\) is any coefficient field for \({\mathfrak O}\), then \(\tilde{\mathfrak O}^ C\) determines the equisingularity class of the generic plane projection of the curve Spec(\({\mathfrak O}{\hat \otimes}_ k\bar k)\); whereas \(\tilde{\mathfrak O}^ Z\) doesn't. reduced local ring; Zariski saturation; Campillo saturation; equisingularity class Núñez, A.: Algebro-geometric properties of saturated rings, J. pure appl. Algebra 59, 201-2014 (1989) Singularities of curves, local rings, Multiplicity theory and related topics Algebro-geometric properties of saturated rings | 0 |
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 behavior of blocks in flat families of finite-dimensional algebras. In a general setting we construct a finite directed graph encoding a stratification of the base scheme according to the block structures of the fibers. This graph can be explicitly obtained when the central characters of simple modules of the generic fiber are known. We show that the block structure of an arbitrary fiber is completely determined by ``atomic'' block structures living on the components of a Weil divisor. As a byproduct, we deduce that the number of blocks of fibers defines a lower semicontinuous function on the base scheme. We furthermore discuss how to obtain information about the simple modules in the blocks by generalizing and establishing several properties of decomposition matrices by \textit{M. Geck} and \textit{R. Rouquier} [Prog. Math. 141, 251--272 (1997; Zbl 0868.20013)]. finite-dimensional algebras; block theory; flat families; representation theory; Brauer reciprocity; decomposition matrices Representations of associative Artinian rings, Classical groups (algebro-geometric aspects), Ring-theoretic aspects of quantum groups, Hecke algebras and their representations, Representations of finite groups of Lie type Blocks in flat families of finite-dimensional algebras | 0 |
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 \(\ell\)-adic local systems on \(\mathbb{A}^1/\overline{\mathbb{F}_p}\) whose images by the monodromy map are given finite groups \(G\) which are generated by their Sylow \(p\)-subgroups. For powers \(q\) of any odd prime \(p\) and any integer \(n\geq 2\), they exhibit explicit local systems, on the affine line \(\mathbb{A}^1\) in characteristic \(p>0\) if \(n\) is even and on the affine plane \(\mathbb{A}^2\) if \(n\) is odd, whose geometric monodromy groups are the finite symplectic groups \(\mathrm{Sp}_{2n}(q)\). For \(n\geq 3\) odd, they show that the explicit rigid local systems on \(\mathbb{A}^1\) in characteristic \(p>0\) constructed in [\textit{N. M. Katz}, Mathematika 64, No. 3, 785--846 (2018; Zbl 1456.11232)] have the special unitary groups \(\mathrm{SU}_n(q)\) as their geometric monodromy groups as conjectured therein, and also prove another conjecture of that paper concerning arithmetic monodromy groups. local systems; monodromy groups; Weil representations; finite symplectic groups; finite unitary groups Structure of families (Picard-Lefschetz, monodromy, etc.), Linear algebraic groups over finite fields Local systems and finite unitary and symplectic groups | 0 |
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 one chooses a generic set \(\{P_1,\dots, P_s\}\) of points in projective space \(\mathbb{P}^n\) over an algebraically closed field \(k\), there is a natural conjecture what the minimal graded free resolution of its vanishing ideal \(I=I(P_1,\dots,P_s) \subseteq S=k[x_0,\dots,x_n]\) should look like. This conjecture is known as the minimal resolution conjecture and was formulated explicitly by \textit{A. Lorenzini} in her Ph.D thesis [see also \textit{A. Lorenzini}, J. Algebra 156, No. 1, 5-35 (1993; Zbl 0811.13008)].
For the case \(n=2\), it was shown to hold by \textit{A. V. Geramita} and \textit{P. Maroscia} [J. Algebra 90, 528-555 (1984; Zbl 0547.14001)], and the case \(n=3\) was solved by \textit{E. Ballico} and \textit{A. V. Geramita} [in: Algebraic geometry, Proc. Conf., Vancouver 1984, CMS Conf. Proc. 6, 1-10 (1986; Zbl 0621.14003)]. Furthermore, the conjecture has been proved in a number of special cases, some of which were settled by computer calculation [cf. \textit{S. Beck} and \textit{M. Kreuzer}, ``How to compute the canonical module of a set of points'', in: Algorithms in algebraic geometry and applications, Proc. MEGA-90 Conf., Santander 1994, Prog. Math. 143, 51-78 (1996)]. Recently, however, D. Eisenbud and S. Popescu have shown that for small numbers of points \(s\leq 2n\), there are infinitely many counterexamples to the minimal resolution conjecture [cf. \textit{D. Eisenbud} and \textit{S. Popescu}, ``Gale duality and free resolutions of ideals of points'' (preprint 1996)].
In the paper under review, the authors demonstrate that the conjecture is true for large numbers of points \(s\gg n\). Unfortunately, as they point out at the end of the paper, their method is not very effective and gives a bound of something like \(s=6^{n^3\log n}\). Nevertheless, their ``methode d'Horace'' is interesting in its own right and has been applied successfully by J. Alexander and the first author in other contexts as well [cf. \textit{J. Alexander} and \textit{A. Hirschowitz}, J. Algebr. Geom. 1, No. 3, 411-426 (1992; Zbl 0784.14001)]. More precisely, the authors reduce the minimal resolution conjecture to the claim that the restriction homomorphism \(H^0 (\mathbb{P}^n, \Omega_{\mathbb{P}^n}^p (\ell))\to \bigoplus_{i=1}^s \Omega_{\mathbb{P}^n}^p (\ell)|_{P_i}\) has maximal rank for \(0\leq p\leq n\) and all \(\ell\geq 0\). This claim is proved via a complicated and subtle induction argument. For the start of this induction the authors cite the thesis of \textit{F. Lauze} [in: ``Sur la resolution des arrangements génériques de points dans les espaces projectifs'' (Thesis Univ. Nice 1994)]. The methode d'Horace is then used to lift maximal rank statements like the desired one from a nonsingular divisor of a smooth projective variety to the variety itself. zero-dimensional scheme; generic set of points in projective space; minimal graded free resolution; minimal resolution conjecture Hirschowitz, A.; Simpson, C., La résolution minimale de l'idéal d'un arrangement général d'un grand nombre de points dans \(\mathbb{P}^n\), Invent. Math., 126, 3, 467-503, (1996) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The minimal resolution of the ideal of a general arrangement of a big number of points in \(\mathbb{P}^ n\) | 0 |
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 \(\alpha\) be a regular local two-dimensional ring, and let \(m = (x, y)\) be its maximal ideal. Let \(m > n > 1\) be coprime integers, and let \(p\) be the integral closure of the ideal \((x^m , y^n )\). Then \(p\) is a simple complete \(m\)-primary ideal, and its value semigroup is generated by \(m, n\).
We construct a minimal system of generators \(\{z_0 ,\dots , z_n \}\) of \(p\), and from this we get a minimal system of generators of the polar ideal \(p'\) of \(p\), consisting of \(n =\theta\) elements. In particular, we show that \(p\) and \(p'\) are monomial ideals. When \(\alpha = k[[x, y]]\), a ring of formal power series over an algebraically closed field \(k\) of characteristic zero, this implies the existence of some relevant property. Greco, S. and Kiyek, K.: The polar ideal of a simple complete ideal having one characteristic pair. Preprint, Politecnico di Torino, Rapporto interno N. 32. Regular local rings, Singularities of curves, local rings Some results on simple complete ideals having one characteristic pair | 0 |
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 \(X\) and \(Y\) be projective varieties in \({\mathbb P}^n_K\). The Samuel multiplicity of an \(\mathfrak m\)-primary ideal in a local ring \((A,{\mathfrak m})\) can be used to define the intersection number of an irreducible component of the intersection of \(X\) and \(Y\). In this paper the authors define a multiplicity sequence \(c_0 (I,A),\dots,c_d (I,A)\) for an arbitrary ideal \(I\) of a \(d\)-dimensional local ring \((A,{\mathfrak m})\) which is closely related to the Stückrad-Vogel intersection cycle. It is defined by means of a certain bigraded ring \(G_{\mathfrak m} (G_I (A))\). The main result of this paper implies that each number of the multiplicity sequence equals the (local) degree of the part of the cycle in a certain dimension. Applications include an interpretation of the Segre classes of a subscheme as multiplicities in a bigraded ring, and a local version of Bezout's theorem. In the case where \(I\) is \(\mathfrak m\)-primary, \(c_0 (I,A)\) is the Samuel multiplicity of \(I\) and it is the only element of the sequence which is non-zero. If the embedded join of \(X\) and \(Y\) has minimal dimension, then again the sequence reduces to only one element different from zero. Interesting connections are made to earlier papers by the authors. intersection theory; Segre classes; Samuel multiplicity; Stückrad-Vogel intersection cycle; embedded join; Bezout's theorem; analytic spread R. Achilles and M. Manaresi, Multiplicities of a bigraded ring and intersection theory, Math. Ann. 309 (1997), 573-591. Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Multiplicity theory and related topics, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Multiplicities of a bigraded ring and intersection theory | 0 |
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 monograph is devoted to the local structure of linear meromorphic differential systems of the form \(du/dz=A(z)u\) in a neighbourhood of an irregular singular point, in the framework of vector bundles and meromorphic connections.
In part I, ``Meromorphic connections and their Stokes phenomena'', the fundamental objects of study are the germs of pairs (V,\(\nabla)\) where V is a holomorphic vector bundle on a disk in the complex plane and \(\nabla\) is a meromorphic connection. The formal aspects of the theory are represented by a functor from the category of germs of pairs to the category of formal differential modules over the corresponding formalizations \({\mathcal F}=C[[z]][z^{-1}];\) the structure theory of formal differential modules is presented in categorical terms, essentially in the form given by P. Deligne.
In Part II a detailed study of the Stokes sheaf and its cohomology is presented and a fundamental result stating that the functor of the first cohomology is representable by an affine space of dimension equal to the irregularity of the endomorphism bundle is proved. Part III is dedicated to the study of loca moduli space for marked and, separately, for unmarked pairs, proving that the analytic complex space corresponding to the above mentioned affine space is a local moduli space for the local isoformal deformations of a pair. A brief historical survey of the main themes and results in the monograph is presented in the Appendix. irregular singular point; formal differential modules; local moduli space; local isoformal deformations; historical survey Babbitt, D.G.; Varadarajan, V.S., Local moduli for meromorphic differential equations, Astérisque, 169-170, 1-217, (1989) Ordinary differential equations in the complex domain, Families, moduli of curves (analytic) Local moduli for meromorphic differential equations | 0 |
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 \({\mathcal{L}}(d;m_1^{s_1}, \dots,m_h^{s_h})\) be the linear system of complex projective plane curves of degree \(d\) having \(s_i\) general points of multiplicity at least \(m_i\) (\(i=1,\dots,h\)). A basic (open) question on these linear systems is that of their dimension. This dimension is well known for nine of fewer points and there are some conjectures (explained in Section 1 of the paper under review) for more than or equal to 10 points. This paper deals with the homogeneous case with ten points, say \({\mathcal{L}}(d;m^{10})\). For this, the conjectures assert that the dimension is the expected one (computed just looking at the number of imposed conditions, cf. Section 1).
The main result in this paper is that for \(d/m \geq 174/55\) the linear system \({\mathcal{L}}(d;m^{10})\) has the expected dimension (see Theorem 0.1), as conjectured. The proof relies on a degeneration argument on the blown-up plane and it seems that, in principle, can be applied beyond the bound \(174/55\), needing the understanding of more complicated degenerations. The authors emphasize that the technique introduced in this paper ``provides the strongest evidence to date for the truth of the conjecture'' because of the absence of theoretical obstructions to carrying it further. linear systems; Segre-Harbourne-Gimigliano conjecture; degeneration techniques Ciliberto, C; Miranda, R, Homogeneous interpolation on ten points, J. Algebraic Geom., 20, 685-726, (2011) Plane and space curves, Divisors, linear systems, invertible sheaves Homogeneous interpolation on ten points | 0 |
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 two-dimensional rational singularity, classical theorems of \textit{J. Lipman} [Publ. Math., Inst. Hautes Étud. Sci. 36, 195--279 (1969; Zbl 0181.48903)] showed that every integrally closed ideal \(I\) satisfies \(I^2=IQ\) for every minimal reduction \(Q\) of \(I\), and that if \(I\), \(J\) are integrally closed ideals, then so is their product \(IJ\). The goal of this paper is to explore the ideal theory for a general two-dimensional normal non-rational singularity.
Let \((A,m)\) be an excellent normal local ring of dimension \(2\) such that \(A\) contains an algebraically closed field \(k\cong A/m\) and let \(f\): \(X\to\mathrm{Spec }A\) be a resolution of singularities with exceptional divisor \(E\). Suppose \(Z\neq 0\) is an anti-nef cycle of \(X\) such that the base locus of the linear system \(H^0(X, O_X(-Z))\) does not contain any component of \(E\). Then the authors show that \(h^1(X, O_X(-Z))\leq p_g(A)=h^1(X, O_X)\). \(Z\) is called a \(p_g\)-cycle if we have equality (and in this case \(O_X(-Z)\) is generated by global sections), and an integrally closed \(m\)-primary ideal \(I\) is called a \(p_g\)-ideal if \(I\) is represented by a \(p_g\)-cycle on some resolution (i.e., \(I=H^0(X, O_X(-Z))\)). If \(A\) is a rational singularity then every anti-nef cycle is a \(p_g\)-cycle and hence every integrally closed ideal is a \(p_g\)-ideal. The authors show that, in general, the class of \(p_g\)-ideal inherits nice properties of integrally closed ideals of rational singularities: for example \(I^2=IQ\) for every minimal reduction \(Q\) of \(I\), and if \(I\), \(J\) are \(p_g\)-ideals then so is their product \(IJ\).
The main result of the paper is that, for \((A,m)\) a two-dimensional normal local ring, there exists a resolution on which \(p_g\)-cycles exist (and thus \(p_g\)-ideals exist). Moreover, they used this to show that if \(A\) is non-regular Gorenstein, then it has good ideals, i.e., \(I^2=IQ\) and \(Q:I=I\). The paper also investigates many other applications of \(p_g\)-ideals, for example the number of minimal generators of certain integrally closed ideals, and (non)existence results on Ulrich ideals for certain elliptic singularity. \(p_g\)-cycles; \(p_g\)-ideals; rational singularities T. Okuma, K-i. Watanabe, and K. Yoshida, Good ideals and \(p_{g}\)-ideals in two-dimensional normal singularities, Manuscripta Math. 150 (2016), 499--520. Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Good ideals and \(p_g\)-ideals in two-dimensional normal singularities | 0 |
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 the paper under review is the study of vector bundles on $\mathbb{P} ^1_A$, where $A$ is a Dedekind ring, done in [\textit{C. C. Hanna}, J. Algebra 52, 322--327 (1978; Zbl 0386.18008)]. In particular it is shown there that all vector bundles on $\mathbb{P} ^1_A$ with $A$ an Euclidean ring have filtrations such that all its factors are line bundles. \par In this paper an algorithm is provided to construct such a filtration for vector bundles of rank $2$ on $\mathbb{P} ^1_A$ with $A$ Euclidian. As a vector bundle on $\mathbb{P} ^1_A$ is defined via an invertible matrix $\sigma $ over $A[x,x^{-1}]$ , the question is reduced to finding matrices $\lambda $ over $A[x]$ and $\rho $ over $A[x^{-1}]$ so that $\lambda \sigma \rho$ is upper triangular. Such an algorithms is provided and examples of how the algorithm operates are given. vector bundle; Euclidean ring; arithmetic surface; projective line; filtration; reduction Vector bundles on curves and their moduli, Euclidean rings and generalizations Construction of a linear filtration for bundles of rank 2 on \(\mathbf{P}^1_{\mathbb{Z}}\) | 0 |
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 \(S\) be an irreducible proper noetherian scheme of finite type over a field \(k\) and assume that \(S\) satisfies Serre's condition \(S_2\). Let \(V\subset S\) be an open subset with closed component of codimension \(\geq 2\) in \(S\) and \(A\) a complete Noetherian local \(k\)-algebra with residue field \(K\). Consider \(X=S\times_k\text{Spec}(A)\), \(U=V\times_k\text{Spec}(A)\), \(X_0=S\times_k \text{Spec}(K)\) and \(U_0=V\times_k \text{Spec}(K)\). Let \(\widehat{X}\) and \(\widehat{U}\) be the formal completions of \(X\) and \(U\) along, respectively, \(X_0\) and \(U_0\) and let \(\mathcal F\) be a vector bundle over \(\widehat U\).
In Section 2 the author proves that \(\mathcal F\) admits always an algebraization if \(\text{codim}_{X_0}(X_0\setminus U_0)\geq 3\) and gives a necessary and sufficient condition for the algebraization of \(\mathcal F\) in the case \(\text{codim}_{X_0}(X_0\setminus U_0)=2\).
In Section 3 the author considers an affine algebraic group \(G\) over \(k\) and shows that, under the condition that the identity component \(G_0\) is reductive, a principal \(G\)-bundle \(\mathcal P\) over \(\widehat U\) admits an algebraization if and only if for a fixed exact representation \(G\hookrightarrow GL(V)\) the associated vector bundle \(\mathcal P_V\) admits an algebraization. In particular, by applying the results in Section 2, \(\mathcal P\) admits an algebraization if \(\text{codim}_{X_0}(X_0\setminus U_0)\geq 3\) and if \(\text{codim}_{X_0}(X_0\setminus U_0)=2\) and the condition of Section 2 holds.
In Section 4 the author provides a categorical restatement of the results in Sections 2 and 3. principal bundles; non-proper scheme; algebraization; formal scheme Baranovsky V.: Algebraization of bundles on non-proper schemes. Trans. Am. Math. Soc. 362, 427--439 (2010) Algebraic moduli problems, moduli of vector bundles Algebraization of bundles on non-proper schemes | 0 |
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 that the algorithm for desingularization of algebraic varieties in characteristic zero of the first two authors is functorial with respect to regular morphisms. For this purpose, we show that, in characteristic zero, a regular morphism with connected affine source can be factored into a smooth morphism, a ground-field extension and a generic-fibre embedding. Every variety of characteristic zero admits a regular morphism to a \(\mathbb{Q}\)-variety. The desingularization algorithm is therefore \(\mathbb{Q}\)-universal or absolute in the sense that it is induced from its restriction to varieties over \(\mathbb{Q}\). As a consequence, for example, the algorithm extends functorially to localizations and Henselizations of varieties. resolution of singularities; functorial; canonical; marked ideal E. Bierstone, P. Milman, and M. Temkin, \( \mathbb{Q}\)-universal desingularization, arXiv preprint 0905.3580v1 (2009). Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Equisingularity (topological and analytic), Global theory of complex singularities; cohomological properties \(Q\)-universal desingularization | 0 |
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 book is devoted to the following fundamental problem of real algebraic geometry: to classify topologically real algebraic sets, to give a topological characterization of all topological spaces which are homeomorphic to real algebraic sets. Main results in this direction achieved at the moment are presented in a comprehensive and self- contained setting. Namely, a complete topological characterization of nonsingular algebraic sets, of algebraic sets with only isolated singularities and of algebraic sets of dimension \(\leq 3\) is given, as well as the affirmative solution to the Nash conjecture that every smooth manifold in \(\mathbb{R}^ n\) is smoothly isotopic to a close nonsingular algebraic set. A detailed introduction to the methods used in this topic is done. In particular, the theory of resolution towers is developed, which is a far generalization of resolution of singularities adapted to the problem. topological classification of real algebraic sets; resolution of singularities; Nash conjecture; resolution towers S. \textsc{Akbulut}\textsc{and} H. \textsc{King}, Topology of Real Algebraic Sets, Mathematical Sciences Research Institute Publications, 25, Springer, New York, 1992. Topology of real algebraic varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Real-analytic manifolds, real-analytic spaces, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes, Stratifications in topological manifolds, Research exposition (monographs, survey articles) pertaining to global analysis, Stratified sets Topology of real algebraic sets | 0 |
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 2-dimensional regular local ring, \(\mathfrak P\) a prime ideal of height 1 and \(u=r/s\) an element of the quotient field of \(R\). Criteria for the existence of an Artinian factor module \(B/X\) such that \({\mathfrak P}=\text{Ann}_R(B/X)\) are formulated and proved by using the ideal \({\mathfrak A}=(r,s)\) and the intersection numbers \(\sigma(\alpha_i,r)\), \(\sigma(\alpha_i,s)\) where \(\alpha_1,\alpha_2,\dots,\alpha_h\) are the branches of \(\mathfrak P\) in the completion \(\widehat R\). coassociated prime ideals; plane curves; intersection numbers; Newton polygon Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Integral closure of commutative rings and ideals, Multiplicity theory and related topics, Singularities of curves, local rings Coassociated prime ideals in 2-dimensional regular rings | 0 |
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 with non-zero identity. For each integer \(i\), Grothendieck defined the \(i\)-th local cohomology of an \(R\)-module \(X\) with respect to an ideal \(\mathfrak {a}\) as follows:
\[\text{H}^i_{\mathfrak {a}}(X):={\underset{n}{\varinjlim}\,\text{Ext}^i_R\left(R/\mathfrak {a}^n, X\right)}.\]
Now, assume \(R\) is a local ring of dimension \(d\) with maximal ideal \(\mathfrak m\). A sequence \(x_1,\dots,x_d\) of elements of \(R\) is called a system of parameters of \(R\) if \(\mathfrak{m}=\sqrt{Rx_1+\cdots+Rx_d}\). In the paper under review, the authors show that when \(4\leq d\), \(2\leq i\leq d-2\) and \(x_1,\ldots,x_i\) is a part of system of parameters of \(R\), then there exist infinitely many prime ideals \(\mathfrak p\) with \(\text{dim}_R(R/\mathfrak p)=i+1\) such that the top local cohomology module \(\text{H}^i_{(x_1,\ldots,x_i)}(R/\mathfrak p)\) with support in \(\{\mathfrak m\}\) is non-Artinian. Artinian module; local cohomology; Noetherian ring Local cohomology and commutative rings, Local cohomology and algebraic geometry, Commutative Noetherian rings and modules Zero-dimensional non-Artinian local cohomology modules | 0 |
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 Es sei \(K\) ein Körper, der über seinem Primkörper endlich erzeugbar ist, und \(V\) das Skelett einer abstrakten arithmetischen Varietät zu \(K\), d.h. die Gesamtheit der lokalen Ringe dieser Varietät (diese gehen durch Quotientenringbildung aus absolut endlich erzeugbaren Teilringen von \(K\) hervor). Kähler schlug eine Definition einer Zetafunktion von \(V\) vor, die sich folgendermaßen schreiben läßt:
(1) \(\zeta(V|s)=\prod_{\mathfrak P}\zeta_{\mathfrak P}(s)\), wobei das Produkt über alle \(O\)-dimensionalen lokalen Ringe \(S\) von \(V\) (bzw. ihre maximalen Primideale \(\mathfrak P\)) läuft und
(2) \(\zeta_{\mathfrak P}(s)=\sum_{\mathfrak g}N(\mathfrak g)^{-s}\) ist (\(\mathfrak g\) jeweils alle Ideale mit endlicher Restklassenanzahl \(N(\mathfrak g)\) von \(S\)). Der Ausdruck (1) läßt sich leicht in eine formale Dirichlet-Reihe umschreiben; ebenso kann man in Verallgemeinerung hiervon Zetafunktionen zu ``Gebilden'' auf \(V\) einführen.
Verf. betrachtet nun den Fall eines Körpers \(K\) der Stufe \(2\) (\(K=\) algebraischer Funktionenkörper von zwei Veränderlichen über einem Galoisfeld oder \(K =\) algebraischer Funktionenkörper von einer Veränderlichen über einem Zahlkörper) unter der Voraussetzung, daß alle \(\mathfrak P\) regulär sind, und bestimmt analytische Eigenschaften von \(\zeta(V|s)\). Hierzu zeigt Verf. im ersten Teil für die Anzahl \(g_\lambda\) der Ideale \(\mathfrak a\) von \(S\) mit fester endlicher Restklassenanzahl \(N(\mathfrak a) = N(\mathfrak p)^{\lambda}\): Die Anzahl \(g_\lambda\) ist ein Polynom vom genauen Grade \(\lambda-1\) in \(N(\mathfrak p)\) mit positiven ganzen Koeffizienten, für welche eine Rekursionsformel hergeleitet wird. Als Hilfüberlegung dient hierbei: Es sei das maximale Ideal \(\mathfrak p\) von \(S\) durch die zwei Elemente \(u,v\) erzeugt, \(b\) sei das von \(v\) in \(S\) erzeugte Ideal, und für zwei Ideale \(\mathfrak a, \mathfrak b\) von \(S\) mit \(\mathfrak a\supset\mathfrak b\) bezeichne \((\frac{\mathfrak b}{\mathfrak a})\) die Länge einer Kompositionsreihe von \(\mathfrak a\) nach \(\mathfrak b\). Sind dann \(\nu_0,\nu_1,\dots,\nu_n,\nu_{n+1},\dots\) ganze Zahlen mit \(\nu_0\geq\nu_1\geq\dots\geq\nu_n>0=\nu_{n+1}=\nu_{n+2}=\dots,\) so ist die Anzahl \(f(\nu_0,\nu_1,\dots)\) der Ideale \(\mathfrak a\) von \(S\) mit
\[
((\mathfrak a+\nu^{i+1})/(\mathfrak a+\nu^i))=\nu_i,~ i=0,1,\dots,
\]
gegeben durch
\[
f(\nu_0,\nu_1,\dots)=N(\mathfrak P)^{\nu_1+\nu_2+\dots+\nu_n}.
\]
Hieraus folgt, daß (2) für \(\Re(s) > 1\) absolut konvergiert zum Werte
\[
(3) \zeta_{\mathfrak P}(s)=\prod_{\rho=0}^{\infty}(1-N(\mathfrak P)^{\rho-(\rho+1)s})^{-1}.
\]
Mit diesem Ausdruck zeigt Verf. durch eine algebraische Zurückführung auf den rationalen Fall (Stufe 1) das folgende Hauptergebnis: Im vorliegenden Fall konvergiert (1) für \(\Re(s) > 2\) absolut gegen eine dort analytische Funktion; das gleiche gilt für die Zetafunktion der ``Gebilde'' auf \(V\).
Für kritische Beispiele zu den analytischen Eigenschaften von \(\zeta(V|s)\) vgl. auch Ref. [Durch Produktdarstellungen erklärte Zetafunktionen. Sammelband Leonhard Euler, Dtsch. Akad. Wiss. Berlin 246--255 (1959; Zbl 0121.04602)]; die unangenehmen Eigenschaften liegen im Wesentlichen daran, daß der lokale Beitrag \(\zeta_{\mathfrak P}(s)\) aus (3) wesentlich komplizierter gebildet ist als der entsprechende bei A. Weil und anderen.
Verf. benutzt die vom üblichen abweichende Kählersche Bezeichnungsweise. number fields; function fields G. Lustig, Über die Zetafunktion einer arithmetischen Mannigfaltigkeit. Math. Nachr.14, 309--330 (1956). Arithmetic varieties and schemes; Arakelov theory; heights, Zeta and \(L\)-functions: analytic theory On the zeta function of an arithmetic manifold. | 0 |
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 rank one local system \(L\) on a smooth complex algebraic variety \(M\) is \textit{admissible} if one has \(H^k(M,L)=H^k(H^*(M), \wedge \alpha)\) for all \(k\), where \(L=\exp(\alpha)\) in the usual sense.
A line arrangement \(A\) is said to be of type \(C_m\) for some \(m \geq 0\) if \(m\) is the minimal number of lines in \(A\) containing all the points of multiplicity at least \(3\).
It was known that any local system \(L\) on the complement \(M\) of a line arrangement of type \(C_m\) for \(m <2\) is admissible.
The authors show that the same result holds for line arrangements of type \(C_2\), and explain clearly why it fails for certain line arrangements of type \(C_3\), as for instance the deleted \(B_3\)-arrangement discovered by A. Suciu. admissible local system; line arrangement; characteristic variety. Nazir, S.; Raza, Z.: Admissible local systems for a class of line arrangements, Proc. am. Math. soc. 137, No. 4, 1307-1313 (2009) Relations with arrangements of hyperplanes, Pencils, nets, webs in algebraic geometry, (Co)homology theory in algebraic geometry, Plane and space curves, Rational and birational maps Admissible local systems for a class of line arrangements | 0 |
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 regular local ring, containing a perfect field \(k\), over which \(R\) is essentially of finite type, and let \(C(R/k)=\{x\in R:\delta (x)=0\) for all \(\delta\in \text{Der}_k (R)\}\). The authors prove that if \(R\) is a semi-local regular excellent and irreducible \(k\)-algebra such that \(R/R^p\) is finite (where \(p=\text{char}(k)>0)\), and \(I\subseteq R\) is an ideal, then there exists an integer \(l\) with the property that if \(x\in R\) with \(\delta(x)\in I^{n+l}\) for all \(\delta\in \text{Der}_k(R)\), then there exists a \(c\in C(R/k)\) with \(x-c\in I^n\). This provides a positive answer to a question raised by C. Huneke [see \textit{R. Fedder}, \textit{C. Huneke} and \textit{R. Hübl}, Proc. Am. Math. Soc. 108, No. 2, 319-325 (1990; Zbl 0691.13028)]. The solution for characteristic 0 was given by \textit{R. Hübl} [Proc. Am. Math. Soc. 127, No. 12, 3503-3511 (1999; Zbl 0938.13008)].
In the characteristic 0 case \(l\) can be bounded by a constant depending only on \(R\). But the situation in the present work is more difficult and the authors are able to find a uniform bound only with restrictions.
The goal is to enable the use of these techniques to clarify or provide alternate proofs of known results and extend work to more general problems. See for example the algebraic formulation of the Kodaria vanishing theorem by \textit{C. Huneke} and \textit{K. Smith} [J. Reine Angew. Math. 484, 127-152 (1997; Zbl 0913.13003)]. tight closure; vanishing theorem; rational singularities; \(F\)-rationality DOI: 10.1007/s002080000095 Modules of differentials, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Vanishing theorems in algebraic geometry Vanishing of differentials along ideals and non-archimedean approximation | 0 |
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 0723.00022.]
The author makes a brief historical note on desingularization: there is an interesting survey of the techniques of \textit{Zariski}, \textit{Abhyankar} and \textit{Hironaka} and a clear summary of the classical proofs of desingularization in dimension 1 and 2.
At the end, there is an announcement of a resolution theorem of dimension 3 singularities of the equation: \(T^ p-f(x_ 1,x_ 2,x_ 3)\) where \(p\) is the characteristic. The author gives some hints about this proof: there are 60 different cases which are controlled with numerical characters built with Newton polyhedras.
Unfortunately, the author did not quote the last results of \textit{E. Bierstone} and \textit{P. D. Milman} [cf. Effective methods in algebraic geometry, Proc. Symp., Castiglioncello 1990, Prog. Math. 94, 11-30 (1991; Zbl 0743.14012) and J. Am. Math. Soc. 2, No. 4, 801-836 (1989; Zbl 0685.32007), of the reviewer in Géométrie algébrique et applications, C. R. 2. Conf. int., La Rabida 1984, Vol. I: Géométrie et calcul algébrique, Trav. Cours 22, 1-21 (1987; Zbl 0621.14015)] and of \textit{O. Villamayor} [``Patching local uniformizations'', Ann. Sci. Éc. Norm. Supér., IV. Sér. 25 (1992)]. desingularization; resolution; dimension 3 singularities; Newton polyhedras Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Development of contemporary mathematics On the resolution of singularities | 0 |
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 paper is devoted to the problem of desingularization of the cotangent sheaf of algebraic varieties and analytic spaces. To be more precise, let \(X_0\) be a reduced algebraic variety over a field of characteristic zero or a complex or real analytic space, and denote by \(\mathrm{Sing } X_0\) the singular locus of \(X_0\). A resolution of singularities of \(X_0\) is, as usual, a proper birational or bimeromorphic morphism \(\sigma: X\to X_0\) from a smooth \(X\) such that \(\sigma\) is an isomorphism over \(X_0\setminus\mathrm{Sing } X_0\) and \(\sigma^{-1}(\mathrm{Sing } X_0)\) is a divisor with simple normal crossings on \(X\). Often one requires also that \(\sigma\) is a composite of blowings-up with smooth admissible (i.e., possessing only normal crossings with exceptional divisor) centers. The main result of the paper states that, for \(X_0\) as above and of dimension \(\dim X_0\leq 3\), there exists a resolution of singularities which is a composite of blowings-up with smooth admissible centers such that the pull-back of the cotangent sheaf of \(X_0\) is locally generated by differential monomials \(d\,(\mathbf{u}^{\mathbf{\alpha}_i})\), \(i=1,\ldots,s\), and \(d\,(\mathbf{u}^{\mathbf{\beta}_j}v_j)\), \(j=1,\ldots,n-s\), where \(n=\dim X_0\), \((u_1,\ldots,u_s,v_1,\ldots,v_{n-s})\) is a local coordinate system on \(X\) such that (1) the support of the exceptional divisor is locally defined by the equation \(u_1\cdots u_s=0\), (2) the multiindices \(\mathbf{\alpha}_1,\ldots,\mathbf{\alpha}_s\in\mathbb{N}^s\) are linearly independent over \(\mathbb{Q}\), and (3) the set \(\{\mathbf{\alpha}_i,\mathbf{\beta}_j\}\) is totally ordered with respect to the componentwise partial ordering of \(\mathbb{N}^s\).
The approach undertaken by the authors of this paper consists in showing that resolution of the cotangent sheaf of a variety \(X_0\) is equivalent to principalization of the logarithmic Fitting ideals \(\mathcal{F}_k(\sigma)\) of the morphism \(\sigma\). These ideals are generated locally by the minors of order \(n-k\) of the logarithmic Jacobian matrix of \(\sigma\) (see Subsection~2.2 of the paper for a precise definition). A resolution \(\sigma: X\to X_0\) is a principalization of the logarithmic Fitting ideals if all \(\mathcal{F}_k(\sigma)\), \(k=0,\ldots,n-1\), are principal and generated locally by monomials in components of the exceptional divisor. The authors prove a general theorem that a principalization of \(\mathcal{F}_k(\sigma)\) at a point \(a\in X\) where the logarithmic Jacobian matrix has rank \(r\) can be achieved by standard resolution technique for \(k=n-1,\ldots,n-r-1\). In dimension \(3\), the only case that remains is to principalize \(\mathcal{F}_1(\sigma)\) at points of logarithmic rank \(0\). The proof in this case is the most difficult part of the paper and perhaps this is the point that cannot be easily generalized to higher dimensions. As in other variants of resolution theorems, the authors construct a technical nonnegative integral invariant that drops after a suitable admissible blowing-up and this allows to proceed by induction. In dimensions \(\dim X_0\geq 4\), the assertion of the main theorem remains a conjecture. resolution of singularities; monomialization; logarithmic differential forms; Fitting ideals Belotto da Silva, A., Bierstone, E., Grandjean, V., Milman, P.: Resolution of singularities of the cotangent sheaf of a singular variety, preprint. arXiv:1504.07280 [math.AG] (2015) Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Complex spaces, Global theory of complex singularities; cohomological properties, Mixed Hodge theory of singular varieties (complex-analytic aspects) Resolution of singularities of the cotangent sheaf of a singular variety | 0 |
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 Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this paper will attempt to fill it.
In one sense, the question can be answered almost completely. Roughly, the space of all possible linearizations is divided into finitely many polyhedral chambers within which the quotient is constant and when a wall between two chambers is crossed, the quotient undergoes a birational transformation which, under mild conditions, is a flip in the sense of Mori. Moreover, there are sheaves of ideals on the two quotients whose blow-ups are both isomorphic to a component of the fibred product of the two quotients over the quotient on the wall. Thus the two quotients are related by a blow-up followed by a blow-down.
The ideal sheaves cannot always be described very explicitly, but there is not much more to say in complete generality. To obtain more concrete results, we require smoothness, and certain conditions on the stabilizers which, though fairly strong, still include many interesting examples. The heart of the paper is devoted to describing the birational transformations between quotients as explicitly as possible under these hypotheses. In the best case the blow-ups turn out to be just the ordinary blow-ups of certain explicit smooth subvarieties, which themselves have the structure of projective bundles.
The last three sections of the paper put this theory into practice, using it to study moduli spaces of points on the line, parabolic bundles on curves, and Bradlow pairs. An important theme is that the structure of each individual quotient is illuminated by understanding the structure of the whole family. So even if there is one especially natural linearization, the problem is still interesting. Indeed, even if the linearization is unique, useful results can be produced by enlarging the variety on which the group acts, so as to create more linearizations. I believe that this problem is essentially elementary in nature, and I have striven to solve it using a minimum of technical machinery. For example, stability and semistability are distinguished as little as possible. Moreover, transcendental methods, choosing a maximal torus, and invoking the numerical criterion are completely avoided. The only technical tool relied on heavily is the marvelous Luna slice theorem. This theorem is used, for example, to give a new, easy proof of the Bialynicki-Birula decomposition theorem. geometric invariant theory; linearization of the group action; flip; blow-up; blow-down; birational transformations between quotients; Luna slice theorem Thaddeus, Michael. \(Geometric invariant theory and flips\). J. Amer. Math. Soc. 9 (1996), no. 3, 691-723. Group actions on varieties or schemes (quotients), Geometric invariant theory, Birational geometry Geometric invariant theory and flips | 0 |
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 invertible sheaf of \({\mathcal O}_ X\)-modules. If \(V\subset \Gamma (X,L)\) is a subvector space of dimension \(r+1,\) it defines a linear system \(| V|\) of effective divisors. The following formula is given for the Jacobian cycle of the linear system \(| V|\) in the smooth quasi-projective \(variety\quad X:\sum^{n-r+1}_{i=0}\left( \begin{matrix} r+i\\ r\end{matrix} \right)c_{n-r+1-i}(\Omega^ 1_ X)c_ 1(L)^ i,\) where \(c_ i\) means the i-th Chern class. linear system; Jacobian cycle; Chern class Cycles and subschemes, Characteristic classes and numbers in differential topology, Divisors, linear systems, invertible sheaves Jacobian loci and determinantal formula | 0 |
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 $I \subset R$ denote an ideal of a regular local ring. The structure of non-zero local cohomology modules $M := H^i_{I}(R)$ is an interesting subject of current research. In particular the following problems became important: (1) $\text{Ass}_R M$ is finite. (2) The Bass numbers of $M$ are finite. (3) $\text{inj.dim}_R M \leq \dim _RM$. (4) $H^i_{\mathfrak{m}}(M)$ is injective.
There are positive answers to these problems in the case of an equi-characteristic regular local ring (see [\textit{C. L. Huneke} and \textit{R. Y. Sharp}, Trans. Am. Math. Soc. 339, No. 2, 765--779 (1993; Zbl 0785.13005)] and [\textit{G. Lyubeznik}, Invent. Math. 113, No. 1, 41--55 (1993; Zbl 0795.13004)]). Moreover, let $R$ be an equi-characteristic local ring presented as $R = S/I$ where $S$ is a regular local ring containing a field. Then Lyubeznik (see [loc. cit.]) defined invariants (nowadays known as Lyubeznik numbers) as certain Betti numbers of $H_I^i(S)$ which are finite by view of (2). By the second and forth author (see [Math. Res. Lett. 20, No. 6, 1125--1143 (2013; Zbl 1311.13022)]) the definition of Lyubeznik numbers was extended to the mixed characteristic case. The major goals of the present paper are:
\begin{itemize} \item[(i)] The relationship between the (standard) Lyubeznik numbers and the mixed characteristic Lyubeznik numbers. \item[(ii)] The investigations when local cohomology satisfy (3) and (4).
\end{itemize}
The main results are: (A) Let $R$ be a finitely generated $\mathbb{Z}$-algebra. Then there is a finite number of primes $W$ such that the (standard) Lyubeznik numbers and the mixed characteristic Lyubeznik coincides for $A$ a localization of $\mathbb{F}_p \otimes_{\mathbb{Z}} R$ and $p \notin W$. (B) Let $I \subset S = \mathbb{Z}[x_1,\ldots,x_n]$ be an ideal such that $\mathbb{Q}\otimes_{\mathbb{Z}}H_I^j(S) \not=0$. Then there is a finite set of primes $W$ such that if $Q \in \text{Supp}_S H^j_I(S)$ not lying over any $p \in W$ then (3) is satisfied for $M = H^j_I(S)_Q$. (C) There is an ideal $I$ of a regular local ring of mixed characteristic $(T,\mathfrak{m})$ such that $M = H^j_I(T)$ fails (3) and (4). local cohomology; Bass numbers; Lyubeznik numbers; injective dimension Local cohomology and algebraic geometry, Local cohomology and commutative rings, Injective and flat modules and ideals in commutative rings Lyubeznik numbers and injective dimension in mixed characteristic | 0 |
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 0741.00064.]
Let \(k\) be an algebraic closed field of characteristic \(p\neq 0\), let \(G\) be a semi-simple simply connected algebraic group, and let \(B\) be a Borel subgroup of \(G\). Let \(n\geq 1\) be an integer, let \(S'\subseteq S\) be two generalized Schubert subschemes of \(G/B^ n\), and let \(\lambda=(\lambda_ 1,\dots,\lambda_ n)\) be an \(n\)-tuple of weights. Set \({\mathcal L}(\lambda)\) be the invertible sheaf on \(G/B^ n\) associated to \(\lambda\), set \({\mathcal K}\) be the kernel of the restriction morphism \({\mathcal L}(\lambda)_{| S}\to{\mathcal L}(\lambda)_{| S'}\). We consider \(M(S,S',\lambda):=H^*(S,{\mathcal K})\) as an element of \({\mathcal D}(B)\) (the derived category of \(B\)-modules) and call it a geometrical complex of \(B\)-modules. Set \(M=M(S,S',\lambda)\), \(FM=M(S,S',p\lambda)\). The group \(B\), the generalized Schubert schemes and the invertible sheaf \({\mathcal L}(\lambda)\) are already defined over \(\mathbb{F}_ p\). Hence the absolute Frobenius map over the natural \(\mathbb{F}_ p\)-forms of these objects induces \(k\)-linear morphisms \(F:M\to FM\) and on the \(B\)- cohomology of these modules. The following theorem is proved. Theorem A: For every integer \(l\), the induced map \(F_ l:H^ l(B,M)\to H^ l(B,FM)\) is injective.
The proof of the theorem uses essentially Kac-Moody data of not finite type. Indeed some embedding of \(G\) into a Kac-Moody group of infinite dimension is used. Note that in order to prove that \({\mathcal O}(B_ n)\) and \({\mathcal O}(C_ n)\) are isomorphic categories, Bernstein and Bien have already similarly used some embeddings of simple Lie algebras into affine Kac-Moody Lie algebras. \(B\)-cohomology; semi-simple simply connected algebraic group; Borel subgroup; generalized Schubert schemes; Frobenius map; Kac-Moody group O. MATHIEU , Frobenius action on the B-cohomology , Proc. of ''Infinite dimensional Lie algebras and groups'' (Luminy 1988 ), ed. V. Kac, World Scientific, Adv. Ser. Math. Phys., 7, 1989 , pp. 39-51. MR 91g:14049 | Zbl 0764.17018 Cohomology of Lie (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Infinite-dimensional Lie groups and their Lie algebras: general properties, Group schemes Frobenius action on the \(B\)-cohomology | 0 |
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 semisimple algebraic group defined over \(\mathbb {Q}_p\), and let \(\Gamma \) be a compact open subgroup of \(G(\mathbb {Q}_p)\). We relate the asymptotic representation theory of \(\Gamma \) and the singularities of the moduli space of \(G\)-local systems on a smooth projective curve, proving new theorems about both:
\begin{itemize}
\item[(1)] We prove that there is a constant \(C\), independent of \(G\), such that the number of \(n\)-dimensional representations of \(\Gamma \) grows slower than \(n^{C}\), confirming a conjecture of \textit{M. Larsen} and \textit{A. Lubotzky} [J. Eur. Math. Soc. (JEMS) 10, No. 2, 351--390 (2008; Zbl 1142.22006)]. In fact, we can take \(C=3\cdot {{\mathrm{dim}}}(E_8)+1=745\). We also prove the same bounds for groups over local fields of large enough characteristic.
\item[(2)]We prove that the coarse moduli space of \(G\)-local systems on a smooth projective curve of genus at least \(\lceil C/2\rceil +1=374\) has rational singularities.
\end{itemize}
For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities. 1 A. Aizenbud and N. Avni, 'Representation growth and rational singularities of the moduli space of local systems', \textit{Invent. Math.}204 (2016) 245-316. MR 3480557. Algebraic moduli problems, moduli of vector bundles, Singularities in algebraic geometry, Asymptotic properties of groups, Linear algebraic groups over local fields and their integers, Deformations of singularities, Symplectic structures of moduli spaces Representation growth and rational singularities of the moduli space of local systems | 0 |
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 how to apply a theorem by Lê Dung Trang and the author about linear families of curves on normal surface singularities to get new results in this area. The main concept used is a precise definition of general elements of an ideal in the local ring of the surface. We make explicit the connection between this notion and the more elementary notion of general element of a linear pencil, through the use of integral closure of ideals . This allows us to prove the invariance of the generic Milnor number (resp. of the multiplicity of the discriminant), between two pencils generating two ideals with the same integral closure (resp. the projections associated). We also show that our theorem, applied in two special cases, on the one hand completes, removing an unnecessary hypothesis, a theorem by J. Snoussi on the limits of tangent hyperplanes, and on the other hand gives an algebraic-constant theorem in linear families of planes curves. Surface singularity; general element; Milnor number; integral closure of ideals; complete ideals; limits of tangent hyperplanes; discriminants Equisingularity (topological and analytic), Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings General elements of an \(m\)-primary ideal on a normal surface singularity | 0 |
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 local system, over the complement of a divisor, underlying a variation of polarizable Hodge structure. We discuss some of the features exhibited in the case of singular divisors with normal crossings by the complexes of \(L_ 2\) and intersection cohomology with coefficients in V. L\({}_ 2\)-cohomology; variation of polarizable Hodge structure; singular divisors; intersection cohomology Cattani, E.; Kaplan, A., \textit{sur la cohomologie \textit{L}_{2} et la cohomologie d'intersection à coefficients dans une variation de structure de Hodge}, C. R. Acad. Sci. Paris Sér. I Math., 300, 351-353, (1985) (Co)homology theory in algebraic geometry, Other homology theories in algebraic topology, Transcendental methods, Hodge theory (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Sur la cohomology \(L_ 2\) et la cohomologie d'intersection à coefficients dans une variation de structure de Hodge. (On the \(L_ 2\)- cohomology and the intersection cohomology with coefficients in a variation of Hodge structures) | 0 |
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 resume the study initiated in our former work [Compos. Math. 154, No. 8, 1659--1697 (2018; Zbl 1403.14097)]. For a generic curve \(C\) in an ample linear system \(| \mathcal{L} |\) on a toric surface \(X\), a vanishing cycle of \(C\) is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of \(C\) to a nodal curve in \(| \mathcal{L} |\). The obstructions that prevent a simple closed curve in \(C\) from being a vanishing cycle are encoded by the adjoint line bundle \(K_X \otimes \mathcal{L} \). In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on \(C\) respectively as an hyperelliptic involution and as a spin structure. In both cases, we determine all the vanishing cycles by investigating the associated monodromy maps, whose target space is the mapping class group \(\text{MCG}(C)\). We show that the image of the monodromy is the subgroup of \(\text{MCG}(C)\) preserving respectively the hyperelliptic involution and the spin structure. The results obtained here support Conjecture 1 in [loc. cit.] aiming to describe all the vanishing cycles for any pair \((X, \mathcal{L})\). mapping class group; Torelli group; spin structures Toric varieties, Newton polyhedra, Okounkov bodies, Other groups related to topology or analysis, Deformations of complex singularities; vanishing cycles The vanishing cycles of curves in toric surfaces. II | 0 |
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 \(J\subset S= K[x_0,\dots, x_n]\) be a monomial strongly stable ideal. The collection \({\mathcal M}f(J)\) of the homogeneous polynomial ideals \(I\), such that the monomials outside \(J\) form a \(K\)-vector basis of \(S/I\), is called a \(J\)-marked family. It can be endowed with a structure of affine scheme, called a \(J\)-marked scheme. For special ideals \(J\), \(J\)-marked schemes provide an open cover of the Hilbert scheme \({\mathcal H}\text{ilb}^n_{p(t)}\), where \(p(t)\) is the Hilbert polynomial of \(S/J\). Those ideals more suitable to this aim are the \(m\)-truncation ideals \(\underline J_{\geq m}\) generated by the monomials of degree \(\geq m\) in a saturated strongly stable monomial ideal \(\underline J\).
Exploiting a characterization of the ideals in \({\mathcal M}f(\underline J_{\geq m})\) in terms of a Buchberger-like criterion, we compute the equations defining the \(\underline J_{\geq m}\)-marked scheme by a new reduction relation, called superminimal reduction, and obtain an embedding of \({\mathcal M}f(\underline J_{\geq m})\) in an affine space of low dimension.
In this setting, explicit computations are achievable in many nontrivial cases. Moreover, for every \(m\), we give a closed embedding \(\phi_m:{\mathcal M}f(\underline J_{\geq m})\hookrightarrow{\mathcal M}f(\underline J_{\geq m+1})\), characterize those \(\phi_m\) that are isomorphisms in terms of the monomial basis of \(\underline J\), especially we characterize the minimum integer \(m_0\) such that \(\phi_m\) is an isomorphism for every \(m\geq m_0\). Hilbert scheme; strongly stable ideal; polynomial reduction relation Bertone, C.; Cioffi, F.; Lella, P.; Roggero, M., Upgraded methods for the effective computation of marked schemes on a strongly stable ideal, J. Symbolic Comput., 50, 263-290, (2013) Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Upgraded methods for the effective computation of marked schemes on a strongly stable ideal | 0 |
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 curve of genus $g$. A coherent system of type $(n,d,n+m)$ is a pair $(E,V)$, where $E$ is a rank $n$ vector bundle on $C$, $\deg (E)=d$ and $V$ is an $(n+m)$-dimensional linear subspace of $H^0(E)$. For any $\alpha >0$ the $\alpha$-slope $\mu _\alpha (E,V) := (d+\alpha k)/n$ gives the notion of $\alpha$-stability for coherent systems on $C$ and a moduli space $G(\alpha;n,d,m)$. When $\alpha$ is near $0$ all sets $G(\alpha;n,d,m)$ are the same and called $S_0(n,d,m)$. Assume that $V$ spans $E$ and call $K$ the kernel of the evaluation map $f: V\otimes \mathcal {O}_C \to E$. $f$ gives an $(m,d,n+m)$-coherent system $D(E,V)$ with $K^\vee$ as the bundle. They state a strong form a conjecture of Butler: if $S_0(n,d,n+m) \ne \emptyset$, then $D(E,V)\in S_0(m,d,m+n)$ for a general $(E,V)\in S_0(n,d,n+m)$ and the rational map $(E,V) \dashrightarrow D(E,V)$ induces a birational morphism between these moduli space. They give some cases in which this conjecture is true and many emptiness or non-emptiness for $S_0(n,d,n+m)$. Some of the results require that $C$ has general moduli. Their strongest results are for $(n,m)=(2,4)$, the first unknown case at the time of the submission. vector bundles on curves; coherent systems; moduli space of coherent systems; stable bundle; Brill-Noether theory of vector bundles; Butler's conjecture Vector bundles on curves and their moduli Generated coherent systems and a conjecture of D. C. Butler | 0 |
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 \(\Lambda\) be a canonical algebra over an algebraically closed base field, and \(d\) the dimension vector of a regular \(\Lambda\)-module. The author studies certain geometric properties of the affine variety \(\text{mod}_\Lambda(d)\), the variety of \(\Lambda\)-modules of dimension vector \(d\). He shows that \(\text{mod}_\Lambda(d)\) is normal if and only if it is irreducible. The variety \(\text{mod}_\Lambda(d)\) is a complete intersection if and only if its dimension equals \(\dim(\text{GL}(d))-\langle d,d\rangle\), where \(\langle-,-\rangle\) stands for the Ringel bilinear form on the Grothendieck group of the category of finite dimensional \(\Lambda\)-modules. Furthermore, the author classifies canonical algebras \(\Lambda\) with the property that \(\text{mod}_\Lambda(d)\) is a complete intersection (respectively normal) for all regular dimension vectors \(d\). It turns out that these properties can be expressed by an inequality for an explicit quantity depending on the lengths of the arms of the Gabriel quiver of \(\Lambda\). canonical algebras; module varieties; complete intersections; normal varieties Bobiński, G., Geometry of regular modules over canonical algebras, Trans. amer. math. soc., 360, 2, 717-742, (2008) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients) Geometry of regular modules over canonical algebras. | 0 |
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 work of \textit{H. Hironaka} in Ann. Math., II. Ser. 79, 109-203 and 203-326 (1964; Zbl 0122.386) there is a deep analysis of the effect of blowing up along a permissible centre; locally this means that the ideal I is a permissible centre for the regular local ring R iff R/I is again regular and moreover the (R/I)-module \(G_ I(R)=\otimes_{n}I^ n/I^{n+1}\) is free (i.e. R is normally flat along I). The paper extends some results well known for permissible blowing ups to pairs (R,I) such that R is normally flat along I but R/I is not necessarily regular. For instance the author obtains some equivalent conditions for R/(x) to be normally flat along P/(x), P being a prime ideal in the regular local ring R of dimension \(d>1\). R/P is not supposed to be regular, but such a hypothesis is replaced by the so called ''isomultiplicity'', a useful tool introduced in the paper: all the elements of any minimal bases of P are required to belong to the same maximal power of \(M=\max imal\) ideal of R. normal flatness; strictly complete intersection; permissible blowing up; isomultiplicity Michela Brundu, Normal flatness and isomultiplicity, Rend. Sem. Mat. Univ. Politec. Torino 40 (1982), no. 1, 163 -- 172 (Italian, with English summary). Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Multiplicity theory and related topics, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Regular local rings Normal flatness and isomultiplicity | 0 |
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 arbitrary characteristic. Let R be the homogeneous coordinate ring of s distinct points \(P_ 1,...,P_ s\) in \({\mathbb{P}}\) \(n_ k\). Let \({\mathfrak O}\) be the localization of R at the origin, which is a Cohen-Macaulay ring of dimension 1. Its type t(\({\mathfrak O})=\dim_ kExt\) \(1_{{\mathfrak O}}(k,{\mathfrak O})\) is denoted by \(t(P_ 1,...,P_ s).\)
Let \(\bar R\) denote the integral closure of R in its total quotient ring, so that \(\bar R=k[T_ 1]\oplus...\oplus k[T_ s]\). Then it holds that (theorem 1.6) \(t(P_ 1,...,P_ s)=s+\ell (\bar R/R)-\ell (Der_ k(\bar R)/Der_ k(R))\). When the points \(P_ i\) are in generic position then the middle term \(\ell (\bar R,R)\) is easy to calculate. Calculation of the last term is the main aim of the paper, and the main result (theorem 2.9) says that, if the points \(P_ i\) are in generic position and \(\binom{n+e-1}{n} <s< \binom{n+e}{n}\) for some \(e\geq 2\), then the term in question is determined by the length \(\ell (Der_ k(R)_{e-1})\) of the degree \(e-1\) part of the graded module \(Der_ k(R)\). This theorem, written in matrix form, gives a matrix which depends only on the homogeneous coordinates of the points and is easy to compute, and whose rank determines \(t(P_ 1,...,P_ s)\). derivation; Cohen-Macaulay type; integral closure of homogeneous coordinate ring of s distinct points Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Morphisms of commutative rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Projective techniques in algebraic geometry Derivations and the Cohen-Macaulay type of points in generic position in n-space | 0 |
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 Noetherian reduced ring. Let \(I\subset R\) be a radical ideal with the following properties:
(1) \(I\) contains a non-zerodivisor of \(R\).
(2) If \(P\subset R\) is a prime such that \(R_P\) ist not normal then \(P\supset I\).
\textit{H. Grauert} and \textit{R. Remmert} [Analytische Stellenalgebren. Unter Mitarbeit von O. Riemenschneider. (1971; Zbl 0231.32001)] proved that \(R\) ist normal if and only if the canonical inclusion \(R\subset \text{Hom}_R(I,I)\) is an equality. This criterion is the basis of an algorithm to compute the normalization, to replace \(R\) by \(\text{Hom}_R(I,I)\) and repeat the procedure if necessary. The ideal \(I\) is called a test ideal for normality. An implementation of this algorithm is available in the computer algebra system \texttt{Singular}. In the paper the number of iterations in terms of localization and completion of the given ring and with respect to the inclusion of test ideals is studied. The algorithm behaves well with respect to localization and completion. It is proved that the number of steps of the global algorithm for \(R\) (using as \(I\) the radical of the singular locus) is at most the maximal number of steps among the strata of a suitable stratification of the singular locus. Equality holds in case that \(R\) is equidimensional and Serre condition \(S_2\) is satisfied. The number of iterations is related to that of the local descendants of the ring. ADE singularities are especially studied. normalization; integral closure; Grauert-Remmert criterion; curve singularity; simple singularity J. Böhm, W. Decker, M. Schulze, Local analysis of Grauert-Remmert-type normalization algorithms. Int. J. Algebra Comput. 24(1), 69-94 (2014) Integral closure of commutative rings and ideals, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities of curves, local rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Local analysis of Grauert-Remmert-type normalization algorithms | 0 |
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{B. Toën} and \textit{M. Vaquié} [J. K-Theory 3, No. 3, 437--500 (2009; Zbl 1177.14022)] initiated a program in relative algebraic geometry starting from a closed monoidal category \(\mathcal{C}\). While doing so, they defined a Zariski topology on the category of commutative monoidal objects of \(\mathcal{C}\) which is the main object of study of the present paper. For \(\mathcal{C}\) certain kinds of closed symmetric monoidal categories generated by their unit, the author proves that for a commutative monoidal object \(A\) of \(\mathcal{C}\), the topological space whose points are prime ideals of \(A\) and whose closed subsets are sets of prime ideals in a locally primitive Gabriel filter (to be defined below), is sober, and its associated locale is Zar(Spec \(A\)) (see [Zbl 1177.14022]), the locale on the category of Zariski open sub-objects Spec \(B \rightarrow\) Spec \(A\) of Spec \(A\) with open immersions as morphisms. The author defines Zariski affine open sub-objects Spec \(B\) of Spec \(A\) to be given by epimorphisms \(A \rightarrow B\) such that \(B\) is finitely presented as an \(A\)-algebra and such that the functor \(-\otimes_A B\) commutes with finite enriched limits. As a consequence of this result, it is also proven that any Zariski affine open sub-object Spec \(B\) of Spec \(A\) has a finite covering by objects Spec \(A_f\). The idea behind this work is that the notion of Zariski open object as a finitely presented flat epimorphism [\textit{A. Grothendieck} and \textit{J. L. Verdier}, Sem. Geom. algebrique Bois-Marie 1963/64, SGA 4, No. 4, Lect. Notes Math. 269, 299--525 (1972; Zbl 0256.18008)] can be extended to the relative setting and that saying Spec \(B\) is a relative Zariski open sub-object of Spec \(A\) is equivalent to having \(B\)-mod as a localization of \(A\)-mod \textit{F. Borceux} and \textit{C. Quinteiro} [Rapp., Sémin. Math., Louvain, Nouv. Sér. 245--260, 171--193 (1996; Zbl 0883.18007)]. Hence the introduction of the category \(\mathcal{B}A\) with one element \(\star\) with End(\(\star)=A\) on which we put a Grothendieck topology called a Gabriel filter on \(A\). An ideal of \(A\) is defined to be an isomorphism class of objects in the category whose objects are monomorphisms \(Z \rightarrow A\) and whose morphisms are monomorphisms in \(A\)-mod\(_{/A}\). The product \(p.p'\) of two ideals \(p\) and \(p'\) of \(A\) is defined to be the image of the morphism \(p \otimes p' \rightarrow A\) and an ideal \(q\) of \(A\) is said to be prime if for any ideals \(p\), \(p'\) such that \(p.p' \subset q\), then either \(p \subset q\) or \(p' \subset q\). Then \(G^q = \cap_{f \in q} G_f\) is called a locally primitive Gabriel filter where \(G_f\) is the Gabriel filter associated to the localization \(A_f\), meaning \(G_f\) is the set of ideals of \(A\) containing a power of \(f\) and this Gabriel filter exists by virtue of the fact that there is a bijection between the set of Gabriel filters of \(A\) and the set of localizations of \(A\)-mod [Zbl 0883.18007]. Essentially then the author proves that the poset of Zariski open sub-objects of Spec \(A\) is the locale associated to the topological space whose points are prime ideals of \(A\) and whose closed sets are \(V(q)=\{p\:|\:q \subset p \}\), \(q\) an ideal of \(A\), and a basis of Zariski open sub-objects in this locale is given by \((\text{Spec} A_f)_{f \in A_0}\), \(A_0=\text{Hom}_{\mathcal{C}}(1, A)\). relative Algebraic Geometry; Zariski topology; Grothendieck topology; locale; localizations Marty, F., Relative Zariski open objects, J. K-Theory, 10, 9-39, (2012) Schemes and morphisms, Local structure of morphisms in algebraic geometry: étale, flat, etc., Monoidal categories (= multiplicative categories) [See also 19D23], Grothendieck topologies and Grothendieck topoi, Enriched categories (over closed or monoidal categories), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Closed categories (closed monoidal and Cartesian closed categories, etc.) Relative Zariski open objects | 0 |
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 Elimination theory is a classical topic in commutative algebra and algebraic geometry. In recent time it became important in context to computational algebra and algebraic geometry. A view towards resultants from a geometric point of view is given by \textit{J. P. Jouanolou} [Adv. Math. 126, No. 2, 119-250 (1997; Zbl 0882.13008); 114, No. 1, 1-174 (1995; Zbl 0882.13007); 90, No. 2, 117-263 (1991; Zbl 0747.13007); 37, 212-238 (1980; Zbl 0527.13005)] and in the book by \textit{I. M. Gelfand, M. M. Kapranov} and \textit{A. N. Zelevinsky} [``Discriminants, resultants and multidimensional determinants'' (Boston 1994; Zbl 0827.14036)]. The main topic of the paper under review is the investigation of aspects of elimination by treating regular sequences and resultants, the authors present in particular elimination theory in weighted projective spaces over arbitrary Noetherian base rings.
The book is divided into four chapters. Chapter I, `Preliminaries', deals with the concept of Kronecker extensions of a ring and its modules and the study of numerical monoids, i.e. submonoids of the non-negative integers \(\mathbb N\) containing almost all elements of \(\mathbb N.\) Note that a weighted polynomial ring over a commutative ring leads to questions about the monoid generated by the weights. Chapter II, `Regular sequences', contains the basic treatment on regular sequences and the concept of (relative) complete intersections and locally complete intersections. A crucial point is the study of generic polynomials. It turns out that a sequence of such polynomials is a regular sequence with a certain additional property of the primary decomposition of the ideal generated by them, described by combinatorial data. Chapter III, `Elimination', presents the main part of elimination theory (with respect to projective spaces). Among others it is shown that the generic elimination ideal is principal. In the case of an integrally closed base ring it follows that the ideal of elimination is divisorial. Here an extended version of duality of graded complete intersection is developed. Chapter IV, `Resultants', is devoted to the study of the resultant for a regular sequence \(F_0,\ldots,F_n\) of homogeneous polynomials in the polynomial algebra \(A[T_0,\ldots,T_n]\) over an integrally closed domain \(A.\) It follows that it is a divisorial ideal (like the elimination ideal) with a more functorial property. Using the Koszul resolution there is a construction of a canonical generator of the resultant ideal. In the classical case this was done in the work of \textit{A. Hurwitz} [Ann. Mat. Pure Appl., III. Ser. 20, 113-151 (1913; JFM 44.0142.02)], the authors' starting point for their generalization for arbitrary Noetherian ground rings. Supplements following each chapter provide extra details and insightful examples.
The exposition provides a complement to sparse elimination theory. In details the difficulties of working over general base rings are carefully investigated. This is essential for many applications, including arithmetic geometry. regular sequences; resultants; elimination; generic polynomials Scheja G., Regular Sequences and Resultants (2001) Linkage, complete intersections and determinantal ideals, Research exposition (monographs, survey articles) pertaining to commutative algebra, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Extension theory of commutative rings, Computational aspects and applications of commutative rings Regular sequences and resultants | 0 |
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 start with a connected complex semi-simple Lie group \(G\) and consider the moduli stack \({\mathcal M}\) of algebraic principal \(G\)-bundle on a smooth complex projective curve \(X\) of genus at least two. The main result is a description of the vector space of jets of regular functions at every point of \({\mathcal M}\). This description is not in terms of the moduli stack but involves certain sheaves on a canonical resolution of cartesian powers of \(X\) such that the preimage of the diagonal divisor has normal crossings.
The first part of the paper is devoted to the combinatorial description of this resolution. The authors use results from and their methods are related to the papers of \textit{H. Esnault}, \textit{V. V. Shehtman}, and \textit{E. Viehweg}, ``Cohomology of local systems on the complement of hyperplanes'' (preprint 1991) and \textit{N. V. Shekhtman} and \textit{A. N. Varchenko} [Invent. Math. 106, No. 1, 139-194 (1991; Zbl 0754.17024)].
The paper under review contains almost no proves. Details and applications are announced to appear elsewhere. moduli stack of algebraic principal bundles on a smooth complex projective curve; normal crossings; N. V. Shekhtman; A. N. Varchenko A. Beilinson, V. Ginzburg, Infinitesimal structure on moduli space of \(G\)-bundles, Intl. Math. Res. Notices 4 (1992), 63-74. Complex-analytic moduli problems, Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Semisimple Lie groups and their representations, Fine and coarse moduli spaces, Divisors, linear systems, invertible sheaves, Formal neighborhoods in algebraic geometry Infinitesimal structure of moduli spaces of \(G\)-bundles | 0 |
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 ring. For a polytope \(P \subset \mathbb R^n\), the polytopal algebra \(R[P]\) is the semigroup ring \(R[S_P]\) of the additive subsemigroup \(S_P\) generated by \(\{(x,1) \mid x \in L_P \}\) where \(L_P = P \cap\mathbb Z^n\). A nonzero element \(v \in \mathbb Z^n\) is called column vector for a polytope \(P\) if there exists a facet \(F \subset P\) such that \(x+v \in P\) whenever \(x \in L_P \setminus F\). \(\text{Col}(P)\) is the set of all column vectors. A column vector determines uniquely a base facet \(F_v\). For a facet \(F \subset P\) and \(z_F \in F\) there exists a unique group homomorphism \(\langle F, -\rangle : \mathbb Z^n \to\mathbb Z\) attaining its minimum on the set \(P \cap\mathbb Z^n\) at the lattice point of \(P\). A polytope \(P\) is balanced if \(\langle P_u, v \rangle \leq 1\) for all \(u,v \in\text{Col}(P)\). If \(u,v \in\text{Col}(P), u + v \neq 0\), and \(x+v \in P_v\) for every \(x \in L_P \setminus P_u\) then \(uv \overset{\text{df}}{=} u + v\) is called product of \(u\) and \(v\). A balanced polytop is \(\text{Col}\)-divisible if its column vectors satisfy: (1) if \(ac\) and \(bd\) exist, and \(a \neq b\), then \(a = db\) or \(b = da\) for some \(d\); (2) if \(ab = cd\) and \(a \neq c\), then there is \(t\) such that \(at = c, td = b\) or \(ct = a, tb = d\). Finally, two polytopes \(P\) and \(P'\) are equivalent if they have the same matrices \((\langle F, v\rangle)\) (resp. \((\langle F', v'\rangle)\)) where \(v\) (resp. \(v'\)) runs over base facets and the partial product structures are the same on \(P\) and \(P'\).
With these notations, the author presents the complete list of all possible equivalence classes of 3-dimensional balanced polytopes. Also a complete list of isomorphism classes of stable elementary automorphism groups \(\mathbb E(R,P)\) for a \(\text{Col}\)-divisible 3-dimensional polytope is given. balanced polytope; \(\text{Col}\)-divisible; elementary automorphism; polytopal linear group; doubling Faramarzi, S. O., Polytopal linear groups and balanced 3-polytopes, Comm. Algebra, 37, 4391-4415, (2009) \(Q\)- and plus-constructions, Miscellaneous applications of \(K\)-theory, Toric varieties, Newton polyhedra, Okounkov bodies, Linear algebraic groups over adèles and other rings and schemes Polytopal linear groups and balanced 3-polytopes | 0 |
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 \(n\), \(a\), \(b\) natural numbers. A matrix pencil \(P\) is given by \(n\) matrices of the same size with coefficients in \(k\), say by \((b \times a)\)-matrices, or, equivalently, by \(n\) linear transformations \(\alpha_i : k^a \rightarrow k^b\) with \(i = 1, \dots, n\). We say that \(P\) is reduced provided the intersection of the kernels of the linear transformations \(\alpha_i\) is zero. If \(P\) is a reduced matrix pencil, a vector \(v \in k^a\) will be called an eigenvector of \(P\) provided the subspace \(\langle \alpha_1(v), \dots, \alpha_n(v) \rangle\) of \(k^b\) generated by the elements \(\alpha_1(v), \dots, \alpha_n(v)\) is \(1\)-dimensional. Eigenvectors are called equivalent provided they are scalar multiples of each other. The set \(\epsilon(P)\) of equivalence classes of eigenvectors of \(P\) is a Zariski closed subset of the projective space \(\mathbb{P}(k^a)\), thus a projective variety. We call it the eigenvector variety of \(P\). The aim of this note is to show that any projective variety arises as an eigenvector variety of some reduced matrix pencil. matrix pencils; eigenvectors; eigenvalues; projective varieties; Kronecker modules; quiver Grassmannians ]] Ringel, C. M., The eigenvector variety of a matrix pencil. Lin. Algebra and Appl. 531 (2017) 447-458, DOI 10.1016/j.laa.2017.05.004 Matrix pencils, Eigenvalues, singular values, and eigenvectors, Families, moduli, classification: algebraic theory, Representations of quivers and partially ordered sets The eigenvector variety of a matrix pencil | 0 |
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 0547.00033.]
Real Clemens structures are presented in such a way we have a trivialisation, with local models, of the neighbourhood of a principal divisor with normal crossings. This is applied to the desingularisation of an algebraically isolated real singularity. Flat forms are found to be preserved by the trivialisation. A generalization of a theorem of Glaeser is also presented. Real Clemens structures Roche, C. A.: Real clemens structures, North-holland math. Stud. 103, 249-270 (1985) Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Differential forms in global analysis, Global theory and resolution of singularities (algebro-geometric aspects), Stratified sets Real Clemens structures | 0 |
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 paper concerns constructing free resolutions of maximal Cohen-Macaulay modules (MCM) over complete intersection rings. These resolutions are often minimal. The results are used to characterize all MCM modules over complete intersections in terms of higher matrix factorizations. \par We provide a bit of historical background and context. The study of MCM modules over Cohen-Macaulay (CM) local rings is a generalization of the representation theory of finite dimensional algebras. In the first interesting case of a hypersurface, Eisenbud in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)] described MCM modules using matrix factorizations. This description arises from a study of the minimal free resolutions of the modules: for example, he showed that an MCM module with no free summands has a periodic resolution of period \(1\) or \(2\) and that these correspond to matrix factorizations of the defining equation. Matrix factorizations have since found applications in algebraic geometry, commutative and homological algebra, representation theory and physics amongst other fields. In [\textit{D. Orlov}, Mat. Contemp. 41, 75--112 (2012; Zbl 1297.14019)], a generalization of the theory of matrix factorizations to complete intersections was initiated, following which, other authors developed it further. However, these do not reveal the structure of minimal free resolutions of MCM modules. On the other hand, although there are many methods to construct free resolutions for wide classes of rings, these rarely yield minimal ones -- the knowledge of infinite minimal free resolutions of modules over rings is sparse. \par In this article (and in [\textit{D. Eisenbud} and \textit{I. Peeva}, Minimal free resolutions over complete intersections. Cham: Springer; (2016; Zbl 1342.13001)), the authors consider the general case of complete intersections. The case of codimension 2 was addressed in \textit{L. L. Avramov} and \textit{R.-O. Buchweitz}, J. Algebra 230, No. 1, 24--67 (2000; Zbl 1011.13007)] and [\textit{D. Eisenbud} and \textit{I. Peeva}, Acta Math. Vietnam. 44, No. 1, 141--157 (2019; Zbl 1419.13023)]. \par We now summarize the results in this paper. Let \(S\) be a Gorenstein local ring and \(M\) a finitely generated CM \(S\)-module of codimension \(c\). Fix a regular sequence \(\underline{f}:=f_1,\dots,f_c\in Ann(M)\) and set \(R:=S/(\underline{f})\). The authors inductively construct \textit{layered} \(S\)-free and \(R\)-free resolutions of \(M\) with respect to \(\underline{f}\). To do this, they use the theory of MCM approximations in the sense of [\textit{M. Auslander} and \textit{R.-O. Buchweitz}, Mém. Soc. Math. Fr., Nouv. Sér. 38, 5--37 (1989; Zbl 0697.13005)]. A brief review of these ideas are provided in section 2. More specifically, they describe codimension one MCM approximations in section 3. Using this, they obtain a \(2\)-term complex of free \(S\)-modules, \(\mathbf{B}^S\), and a map of complexes, \(\psi_{\bullet}^S:\mathbf{B}^S[-1]\rightarrow \mathbf{L'}\), where \(\mathbf{L'}\) is the inductively obtained \(S\)-free layered resolution of the essential MCM approximation of \(M\) over \(R':=R/(f_1,\dots,f_{c-1})\). In section \(4\), the layered \(S\)-free resolution of \(M\) is then constructed as the mapping cone of the map induced by \(\psi_{\bullet}^S\) between \(\mathbf{K}\otimes_S\mathbf{B}^S[-1]\rightarrow \mathbf{L'}\), where \(\mathbf{K}\) is the Koszul complex resolving \(R'\) over \(S\). For the \(R\)-free resolutions, the authors additionally make use of complete intersection (CI) operators and the Shamash construction -- these are reviewed in section \(5\). Let \(\mathbf{L'}\) now denote the inductively obtained \(R'\)-free layered resolution of the essential MCM approximation of \(M\) over \(R'\). Using the codimension one MCM approximations in section 3, a \(2\)-term complex of \(R'\)-free modules, \(\mathbf{B}\), and a map of complexes \(\psi_{\bullet}:\mathbf{B}[-1]\rightarrow \mathbf{L'}\) is obtained. In section \(6\), the \(R\)-free layered resolution of \(M\) is then constructed as the Shamash complex of the mapping cone of \(\psi_{\bullet}\). Note here that the mapping cone of \(\psi_{\bullet}\) is a \(R'\)-free resolution of \(M\). For high \(R\)-syzygies of a given \(R\)-module \(N\) of finite projective dimension over \(S\), these resolutions coincide with the ones constructed in [\textit{D. Eisenbud} and \textit{I. Peeva}, Minimal free resolutions over complete intersections. Cham: Springer (2016; Zbl 1342.13001)]. An alternate approach to the construction of the layered \(R\)-free resolutions is presented in section 9. It is obtained from a periodic exact sequence of \(R\)-modules that generalizes the \(R\)-free periodic resolution of a module over a hypersurface constructed in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]. The ``layered'' terminology comes from the fact that these resolutions come with a natural filtration by subcomplexes, whose subquotients are the layers. \par Sections \(7\) and \(8\) concern the minimality of the layered resolutions constructed. In section \(7\), a criteria for minimality is provided in terms of the injectivity of certain CI operators (Theorem 7.1). This is then used to show under mild hypothesis that if \(N\) is a finitely generated MCM \(R\)-module of finite projective dimension over \(S\), then the layered resolutions over \(R\) and \(S\) with respect to \(\underline{f}\) of the \(n\)-th syzygy of \(M\) over \(R\) are minimal when \(n\geq 3+\text{max}\{c-2,r(\underline{f},N)\}\). Here \(r(\underline{f},N)\) is a function of the Castelnuovo-Mumford regularities over the rings of CI operators corresponding to the \(f_i\) of Ext modules involving the essential MCM approximations of \(N\) with respect to the \(S/(f_1,\dots,f_i)\). In particular, when \(S\) has an infinite residue field, the layered resolutions of sufficiently high \(R\)-syzygies of a given \(R\)-module \(N\) are minimal. Questions relating to the invariant \(r(\underline{f},N)\) are also explored in section \(8\). \par In section 10, an inductive definition of a CI matrix factorization is given (Definition 10.2) and is then used to define a CI matrix factorization module. The definitions here are essentially equivalent to the ones introduced by the authors in [\textit{D. Eisenbud} and \textit{I. Peeva}, Minimal free resolutions over complete intersections. Cham: Springer (2016; Zbl 1342.13001)]. These are then used to characterize all MCM modules over a complete intersection, generalizing the analogous result for hypersurfaces in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]: A finitely generated \(R\)-module \(N\) is MCM if and only if it is a CI matrix factorization module for the sequence \(\underline{f}\). free resolutions; complete intersections; CI operators; Eisenbud operators; maximal Cohen-Macaulay modules Syzygies, resolutions, complexes and commutative rings, Cohen-Macaulay modules, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Complete intersections Layered resolutions of Cohen-Macaulay modules | 0 |
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 system of \(n\) polynomial equations in \(n\) variables over an algebraically closed field \(K\), where the occurring monomials are fixed and the coefficients are ``generic''. In this paper formulae for the number of roots of such a system in any \(K^*\)-stable subset of the affine space \(K^n\) are given. There are two main results, a recursion over ``shifted'' and lower-dimensional systems (main theorem I), and an expression for a special situation in terms of mixed volumes (affine point theorem II). The paper also contains a characterization of ``genericity'' of the coefficients in terms of sparse resultants (main theorem II). Moreover, for the case of affine space minus an arbitrary union of coordinate hyperplanes, the formulae yield upper bounds on the number of possible isolated roots. With some examples, the author illustrates that those bounds are sharper than the ones obtained from previously known results. There are two main new ideas, one is to construct from the Newton-polytopes of the given system a suitable projective toric compactification for \(K^n\), and the other is to consider ``shifts'' of the modified polytopes to take into account roots in certain coordinate subspaces. counting solutions of polynomial equations; mixed volumes Rojas, J. M.: Toric intersection theory for affine root counting. J. pure appl. Algebra 136, 67-100 (1999) Enumerative problems (combinatorial problems) in algebraic geometry, Mixed volumes and related topics in convex geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Numerical computation of solutions to systems of equations Toric intersection theory for affine root counting | 0 |
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 rational map from \(\mathbb P^n\) to itself can be described by \(n+1\) homogeneous polynomials of degree \(d\) in \(n+1\) variables, so coefficients of the polynomials define a point on \(\mathbb P^N\) for \(N+1 = (n+1)\binom {n+d}d\). Within this \(\mathbb P^N\), the set of morphisms forms an open affine variety \(\mathrm{Hom}_d^n\). Choosing a different coordinate on \(\mathbb P^n\) amounts to a conjugation by an element of \(\mathrm{PGL}(n+1)\), so the natural moduli space for dynamical systems on \(\mathbb P^n\) is the quotient of \(\mathrm{Hom}_d^n\) by \(\mathrm{PGL}(n+1)\). \textit{C. Petsche, L. Szpiro}, and \textit{M. Tepper} [J. Algebra 322, No. 9, 3345--3365 (2009; Zbl 1190.14013)] have shown that this exists as a geometric quotient in the sense of Geometric Invariant Theory, generalizing the result of \textit{J. Silverman} [Duke Math. J. 94, No. 1, 41--77 (1998; Zbl 0966.14031)] for \(n=1\). There is also a previous work by the author [Acta Arith. 146, No. 1, 13--31 (2011; Zbl 1285.37020)], explicitly describing stable and semistable loci and giving a bound on on the stabilizer group of a point in the moduli space.
In this article, the author restates semistable reduction theorem for a closely-related moduli space given by the quotient of \(\mathrm{Hom}_d^n\) by \(\mathrm{SL}(n+1)\), and further proves some refinements. The semistable reduction theorem is a standard result in GIT, stating that the semistability on the generic fiber implies that on the special fiber, possibly after taking a finite cover. In the current context, this means that given a complete curve \(C\) and a semistable map \(\varphi\) on \(\mathbb P^n_{K(C)}\), there exist an abstract curve \(D\) mapping finite-to-one onto \(C\) and a self-map \(\Phi\) on a \(\mathbb P^n\)-bundle on \(D\) such that \(\Phi\) is equivalent to the pullback of \(\varphi\) to \(D\) upon some coordinate changes and the reduction of \(\Phi\) at each point of \(D\) is semistable. In particular, one can apply this theorem to a curve \(C\) contained inside the semistable part \(M_d^{n,ss}\) of the moduli space, since each point of \(C\) represents a semistable self-map of \(\mathbb P^n\). The author then answers several natural questions arising from this setup. First, he proves that for any \(n\) and \(d\), there exists a curve in \(M_d^{n,ss}\) such that no trivial bundle on a finite cover satisfies the semistable reduction. As a result, such a curve does not have a complete curve in \(\mathrm{Hom}_d^{n,ss}\) as a finite cover. Secondly, the author shows that an infinitely many non-isomorphic bundle classes can occur for some rational curves in \(M_d^{n,ss}\). Thirdly, the author proves that if the trivial bundle on \(D\) satisfies semistable reduction, the degree of the cover \(D\to C\) is bounded in terms of the sizes of the stabilizer groups for the points on \(D\). The proofs involve standard GIT arguments together with explicit analyses of the closures of \(\mathrm{PGL}(n+1)\)-orbits of some polynomial families. GIT; semistable reduction; moduli space; dynamical system [10]A. Levy, The semistable reduction problem for the space of morphisms on Pn, Algebra Number Theory 6 (2012), 1483--1501. Geometric invariant theory, Families and moduli spaces in arithmetic and non-Archimedean dynamical systems, Arithmetic dynamics on general algebraic varieties The semistable reduction problem for the space of morphisms on \(\mathbb P^n\) | 0 |
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 Dedekind domain with field of fractions \(K\) and \(A\) a central simple algebra over \(K\). If \(\Lambda\) is a hereditary \(R\)-order of \(A\), then Artin canonically associated to it its so-called Severi-Brauer scheme \(X\), by letting \(X(R')\) consist for any commutative \(R\)-algebra \(R'\) of all left ideals \(I'\) of \(\Lambda'=\Lambda\otimes R'\), such that \(\Lambda'/I'\) is projective over \(R'\) and \(I'\) has rank \(n\) over \(R'\). In this case, the generic fibre \(X_K\) of \(X\) over \(\text{Spec}(R)\) is the classical Severi-Brauer variety associated to the central simple algebra \(A\). On the other hand, for any closed point \(p\) of \(\text{Spec}(R)\) the special fibre \(X_p\) is the scheme of left ideals of rank \(n\) of \(\Lambda/p\Lambda\). If \(X_p\) is non-degenerate, i.e., if \(\Lambda\) is unramified over \(p\), then \(\Lambda/p\Lambda\) is a central simple algebra over \(k(p)\), the residue field of \(p\), and \(X_p\) is a Severi-Brauer variety as well.
The principal purpose of this paper is to study the Chow groups of dimension 0 of the special fibres of Artin's models, i.e., the connected component containing the generic fibre of the previously defined Severi-Brauer schemes.
The main result is as follows. Let \(R\) be a complete discrete valuation ring with maximal ideal \(pR\) and field of fractions \(K\), and with perfect residue field. Let \(X\) be an Artin model over \(\text{Spec}(R)\) associated to a central simple algebra \(A\). If (a) the index of \(A\) has no square factors and \(A\) is not ramified or (b) \(A\) is totally ramified (of arbitrary index), then \(A_0(X) =0\). Of course, this applies to the case where \(k\) has cohomological dimension (at most) 1, for in this case \(A\) is totally ramified, as the skew residue field of \(A\) is trivial. hereditary orders; central simple algebras; Severi-Brauer schemes; Severi-Brauer varieties; Chow groups; Artin models Emmanuelle Frossard, Fibres dégénérées des schémas de Severi-Brauer d'ordres, J. Algebra 198 (1997), no. 2, 362 -- 387 (French). Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Finite-dimensional division rings, Algebraic cycles Degenerated fibers of Severi-Brauer schemes of orders | 0 |
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 Gelfand, Graev, Kapranov, and Zelevinsky defined certain linear systems of partial differential equations, now known as \(A\)-\textit{hypergeometric} or \textit{GKZ hypergeometric} systems \(H_A(\beta)\), whose solutions generalize the classical hypergeometric series. These holonomic systems are constructed from discrete input consisting of an integer \(d \times n\) matrix \(A\) along with continuous input consisting of a complex vector \(\beta\in\mathbb C^d\). Assume that the convex hull conv\((A)\) of the columns of \(A\) does not contain the origin. The matrix \(A\) determines a semigroup ring \({\mathbb C}[\mathbb NA]\), and the dimension rank\((H_A(\beta))\) of the space of analytic solutions of \(H_A(\beta)\) is independent of \(\beta\) whenever \({\mathbb C}[\mathbb NA]\) is Cohen-Macaulay. In this note, the authors use the combinatorics of \(\mathbb Z^d\)-graded local cohomology to characterize the set of parameters \(\beta\) for which the rank goes up, in the simplicial case. The premise is the standard fact that a semigroup ring \({\mathbb C}[\mathbb NA]\) fails to be Cohen-Macaulay iff a local cohomology module \(H_{\mathfrak m}^i({\mathbb C}[\mathbb NA])\) is nonzero for some cohomological index \(i\) strictly less than the dimension \(d\) of \({\mathbb C}[\mathbb NA]\). After gathering some facts about \(A\)-hypergeometric systems, the authors prove the simplicial case of the following: Assume that conv\((A)\) has dimension \(d-1\). The set of parameters \(\beta\in \mathbb C^d\) such that rank\((H_A(\beta))\) is greater than the generic rank equals the Zariski closure (in \(\mathbb C^d\)) of the set of \(\mathbb Z^d\)-graded degrees where the local cohomology \(\bigoplus_{i<d}H_{\mathfrak m}^i({\mathbb C}[\mathbb NA])\) is nonzero. Using a different approach, the authors prove the full conjecture in the paper \textit{L. F. Matusevich, E. Miller}, and \textit{U. Walther} [Homological methods for hypergeometric families, J. Am. Math. Soc. 18, No. 4, 919-941 (2005; Zbl 1095.13033)]. \(A\)-hypergeometric systems; simplicial; combinatorics of \(\mathbb Z^d\)-graded local cohomology; local cohomology for semigroup rings; \(A\)-hypergeometric module; local cohomology module; rank-jumping parameter Laura Felicia Matusevich and Ezra Miller, Combinatorics of rank jumps in simplicial hypergeometric systems, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1375 -- 1381. Other hypergeometric functions and integrals in several variables, Toric varieties, Newton polyhedra, Okounkov bodies, Commutative rings of differential operators and their modules, Local cohomology and commutative rings, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Cohen-Macaulay modules, Ordinary and skew polynomial rings and semigroup rings, Semigroup rings, multiplicative semigroups of rings Combinatorics of rank jumps in simplicial hypergeometric systems | 0 |
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 \({\mathfrak O}=G\cdot e\) be the adjoint orbit of a nilpotent element \(e\) in the Lie algebra \(\mathfrak g\) of a complex connected semi-simple Lie group \(G\). The Zariski closure \(\bar {\mathfrak O} \) is an affine algebraic variety. The author proves that the normalisation \(\bar {\mathfrak O}^{\text{norm}}\) of \(\bar{\mathfrak O}\) is Gorenstein and has only rational singularities.
He uses general duality theory to prove the key lemma: Let \(\pi \colon X\to Y\) be proper birational, with \(X\) smooth and \(Y\) normal. Suppose there is a morphism \(\phi\colon {\mathcal O}_ X \to \omega_ X\), inducing an isomorphism \(\pi_ *\phi \colon \pi_ *{\mathcal O}_ X \to \pi_ *\omega_ X\). Then \(Y\) is Gorenstein with rational singularities.
The key lemma is applied to a desingularisation \(X\) with \(G\)-action of \(\bar {\mathfrak O}^{\text{norm}}\); the morphism comes from a \(G\)-invariant section of \(H^ 0(X,\omega_ X)\). The last section gives a direct construction of this section, due to \textit{D. I. Panyushev}, who also proved the main result [see Funct. Anal. Appl. 25, No. 3, 225-226 (1991); translation from Funkts. Anal. Prilozh. 25, No. 3, 76-78 (1991; Zbl 0749.14030)]. Gorensteinness; desingularisation [H1] V. Hinich,Onthe singularities of nilpotent orbits, Israel Journal of Mathematics73 (1991), 297--308. Complete intersections, Global theory and resolution of singularities (algebro-geometric aspects), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) On the singularities of nilpotent orbits | 0 |
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 Minimal free resolution of ideals are a powerful tool, encoding a significant amount of information about the ideal. Given a monomial ideal $I$ in a polynomial ring $R$, there are a variety of results that associate to a resolution a combinatorial or geometric object. In this paper, the focus is on CW-complexes, working over a field of characteristic $p>0$. Given a minimial free resolution \[ \mathcal{F}: 0 \leftarrow F_0 \leftarrow R_1 \leftarrow \cdots \leftarrow F_n \leftarrow 0 \] of a monomial ideal $I$, a CW-complex supporting $\mathcal{F}$ is one for which there is a bijection between the basis elements of $F_i$ and the $i-$cells of the complex that preserves the key properties when passing from the resolution maps to the corresponding boundary maps of the CW complex. It has been shown that there exist monomial ideals whose resolutions are not supported by a CW-complex. An alternate way to represent minimal free resolutions is to use posets. It was shown in [\textit{E. Batzies} and \textit{V. Welker}, J. Reine Angew. Math. 543, 147--168 (2002; Zbl 1055.13012)] that for any monomial ideal $I$, there is a poset supporting its free resolution. The main result of this paper shows that if the minimal resolution $\mathcal{F}$ of a monomial ideal $I$ is suppored by a CW-complex $X$ whose $2$-skeleton is regular, there there is a CW-complex $Y$ that also supports $\mathcal{F}$ for which the incidence poset of $Y$ supports $\mathcal{F}$ in the sense of \textit{T. B. P. Clark} and \textit{A. B. Tchernev} [Trans. Am. Math. Soc. 371, No. 6, 3995--4027 (2019; Zbl 1471.13030)]. commutative algebra; CW-resolutions; minimal free resolutions; monomial ideals Syzygies, resolutions, complexes and commutative rings, Combinatorial aspects of commutative algebra, Chain complexes in algebraic topology, Projective and free modules and ideals in commutative rings, Homotopy theory and fundamental groups in algebraic geometry, Combinatorial aspects of simplicial complexes CW-resolutions of monomial ideals that are supported on face posets | 0 |
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 simply connected semisimple algebraic group over an algebraically closed field of characteristic zero and \(V(\lambda)\) be a simple \(G\)-module of highest weight \(\lambda\). Then \(\text{End}V(\lambda)\) is a simple \((G\times G)\)-module and \(\mathbb{P}(\mathrm{End}V(\lambda))\) is a \((G\times G)\)-variety. The orbit closure \(X_{\lambda}=\overline{(G\times G)[{\roman{id}}_{V(\lambda)}]}\subset\mathbb{P}(\text{End}V(\lambda))\) is an equivariant compactification of the image of \(G\) in \(\mathrm{PGL}(V(\lambda))\) considered as a symmetric space. The nature of singularities of the varieties \(X_{\lambda}\) is studied in the paper under review.
The following theorems are the main results of the paper: (A) \(X_{\lambda}\) is normal if and only if, whenever the support \(\text{Supp}(\lambda)\) of the highest weight \(\lambda\) contains a long simple root in a non-simply laced component of the Dynkin diagram of \(G\), it contains the short simple root which is adjacent to a long one in the component; (B) \(X_{\lambda}\) is smooth if and only if it is normal and the following three conditions hold:
(1) the intersection of \(\text{Supp}(\lambda)\) with each component of the Dynkin diagram is connected and is an extreme vertex of the component whenever it is a one-element set;
(2) \(\text{Supp}(\lambda)\) contains every junction vertex together with at least two neighboring vertices;
(3) the complement of \(\text{Supp}(\lambda)\) is a subdiagram having all components of type A.
These results specify general criteria for normality and smoothness of projective compactifications of reductive groups obtained by \textit{D. A. Timashev} [Sb. Math. 194, No. 4, 589--616 (2003); translation from Mat. Sb. 194, No. 4, 119--146 (2003; Zbl 1074.14043)]. A criterion for projective normality of \(X_{\lambda}\) was obtained by \textit{S. S. Kannan} [Math. Z. 239, No. 4, 673--682 (2002; Zbl 0997.14012)]: \(X_{\lambda}\) is projectively normal if and only if \(\lambda\) is minuscule. It should be noted that the criteria obtained in the paper are of pure combinatorial nature and easy to check. In fact, Theorems A and B are extended in the paper to a wider class of simple projective group compactifications \(X_{\Sigma}=\overline{(G\times G)[{\roman{id}}_V]}\subset\mathbb{P}(\text{End}V)\), where \(V=\bigoplus_{\lambda\in\Sigma}V(\lambda)\) is any (multiplicity-free) \(G\)-module and the set of highest weights \(\Sigma\) contains a unique maximal element \(\lambda_{\max}\) with respect to the dominance order. (``Simple'' means here ``having a unique closed orbit''.) In particular, it follows that \(X_{\Sigma}\) is smooth if and only if \(X_{\lambda_{\max}}\) is smooth.
Methods of the proofs include using wonderful compactifications of semisimple groups, results of Kannan on projective normality, Timashev's criterion for smoothness, etc. Further generalizations to completions of arbitrary symmetric spaces \(G/H\) in \(\mathbb{P}(V(\lambda))\) are also possible, but require different methods. semisimple algebraic groups; projective representations; group compactifications; wonderful varieties; symmetric spaces Bravi, Paolo; Gandini, Jacopo; Maffei, Andrea; Ruzzi, Alessandro, Normality and non-normality of group compactifications in simple projective spaces, Ann. Inst. Fourier (Grenoble), 0373-0956, 61, 6, 2435\textendash 2461 (2012) pp., (2011) Compactifications; symmetric and spherical varieties, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations Normality and non-normality of group compactifications in simple projective spaces | 0 |
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 Hiroshima Math. J. 17, 47-66 (1987; Zbl 0625.13015) and 361-372 (1987; Zbl 0639.13016), the author introduced the notion of various genera of a noetherian local ring. The aim of this paper is to study these genera, especially the normal arithmetic genera, in the case of one-dimensional local rings (or curve singularities). First, we consider the glueings of a finite number of local rings and introduce the notion of local rings which are good crossings. Especially, we examine the genera of local rings which are obtained by glueings. Then we give various characterizations and structure theorems for curve singularities with normal arithmetic genera zero or one.'' arithmetic genera; one-dimensional local rings; curve singularities A. Ooishi, Genera of curve singularities, to appear. Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities in algebraic geometry Genera of curve singularities | 0 |
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 problem of characterizing the possible Hilbert functions of local Artinian level rings of codimension two. The central theorem in the paper provides, given an ``admissible'' numerical function \(h\), an effective method to produce a zero-dimensional ideal of \(k[[x,y]]\) such that \(R/I\) is a local Artinian level ring having \(h\) as a Hilbert function. As a consequence, the author obtains the main result of the paper, which gives a characterization of the possible Hilbert functions of codimension two local Artinian level rings of given type and socle degree. The techniques used in the paper include the use of a similar notion to that of Gröbner bases in the formal power series ring, namely enhanced standard bases, for which the author gives a characterization in the case of ideals of height two. The author uses also the good properties of the generic initial ideal (Gin) of ideals in \(k[[x_1,\dots, x_n]]\).
The results in the paper extend previous work by \textit{J. Briançon} [Invent. Math. 41, 45--89 (1977; Zbl 0353.14004)], \textit{A. Iarrobino} [Proc. Symp. Pure Math. 40, Part 1, 593--608 (1983; Zbl 0563.13010), Lect. Notes Math. 1124, 146--165 (1985; Zbl 0567.14001), J. Algebra 272, No. 2, 530--580 (2004; Zbl 1119.13015)] and \textit{J. V. Chipalkatti} and \textit{A. V. Geramita} [Mich. Math. J. 51, No. 1, 187--207 (2003; Zbl 1097.13514)]. Hilbert functions; level local artinian rings; monomial standard bases; minimal number of generators; generic initial ideals Bertella, V, Hilbert function of local Artinian level rings in codimension 2, J. Algebra, 321, 1429-1442, (2009) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Complete intersections, Commutative Artinian rings and modules, finite-dimensional algebras, Polynomial rings and ideals; rings of integer-valued polynomials, Formal power series rings Hilbert function of local Artinian level rings in codimension two | 0 |
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 0563.00006.]
\({\mathcal O}_ V(K)\) being the canonical sheaf of a variety V over \({\mathbb{C}}\), the canonical ring of V is defined as the graded ring R of sections of \({\mathcal O}_ V(K)^{\otimes m}\), \(m=0,1,... \). The variety V is said to be free of general type if R contains homogeneous elements \(\eta_ 0,...,\eta_ n\) \((n=\dim V)\) which become nowhere simultaneously zero and if moreover the selfintersection \(K^ n>0\). Pluriregular means \(H^ i(V,{\mathcal O}_ V)=0\) for \(i=1,...,n-1\). The canonical model of V is \(X=\Pr oj(R)\). The author extends his method of quasi-generic projections \(\psi: X\to \Sigma,\) which he used to classify regular surfaces \((n=2)\) birationally [cf. Algebraic geometry, Proc. Internat. Conf., Bucharest/Rom. 1982, Lect. Notes Math. 1056, 68-111 (1984; Zbl 0557.14019)], to higher dimensional varieties, assumed to be pluriregular and free of general type. By definition, \(\Sigma\) is a hypersurface and \(\psi\) is either birational or of degree 2 with \(\Sigma\) normal. The hypersurfaces X are defined by equations expressing that the determinant of a symmetric matrix of homogeneous polynomials is zero, the set of these matrices being defined by a rank condition. higher dimensional varieties free of general type; pluriregular; hypersurfaces; classification; determinantal variety; canonical ring Catanese, F.: Equations of pluriregular varieties of general type. Progr. math. 60, 47-67 (1985) Moduli, classification: analytic theory; relations with modular forms, Determinantal varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Equations of pluriregular varieties of general type | 0 |
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 \(p_1,\dots, p_n\) be \(n\) distinct points in the affine plane \(A^2_{\mathbf k}\) with no two points lying on the same vertical line. Let \(m_{ij}\in\mathbf k\) be the slope of the unique line joining \(p_i\) and \(p_j\). Thus \((m_{12},\dots,m_{n-1,n})\) is a point in affine space \(A^s_{\mathbf k}\) with \(s=\frac{n(n-1)}{2}\). The affine slope variety \(P(n)\) is defined as the closure of the locus of all such points arising from the data \((p_1,\dots,p_n)\). Note that in [Trans. Am. Math. Soc. 355, 4151--4169 (2003; Zbl 1029.05040)] the author considered the affine slope variety \(P(G)\) defined by a graph \(G\), and the variety \(P(n)=P(K_n)\) is defined by the complete graph \(K_n\) in these terms.
The affine slope variety turns out to have an unexpectedly rich combinatorial and geometric structure. In order to investigate its properties the author uses combinatorics (graph theory and recursive enumeration of trees), commutative algebra (Gröbner bases and Stanley-Reisner theory), and algebraic geometry.
Let us formulate the main results of the article. Let \(R_n=\mathbf k[m_{12},\dots,m_{n-1,n}]\) and \(I_n\) be the ideal generated by the tree polynomials of all rigidity circuits in the complete graph \(K_n\). Then the variety \(P(n)\) is defined scheme-theoretically by \(I_n\). Moreover, the tree polynomials of the so-called wheel subgraphs of \(K_n\) already generate \(I_n\) and form a Gröbner basis with respect to a certain graded lexicographic order. The dimension of \(P(n)\) equals \(2n-3\), and explicit formulas for the degree of \(P(n)\) and for the Hilbert series of \(R_n/I_n\) in terms of perfect matchings on the set \(\{1,2,\dots,2n-4\}\) are given. It is also shown that the ring \(R_n/I_n\) is Cohen-Macaulay.
Other spaces related to graph varieties include the Fulton-Macpherson compactification of configuration space and the De Concini-Procesi wonderful model of subspace arrangements. The results of this article may serve as a starting point for studying these relations in more detail. affine slope variety; Gröbner basis; Stanley-Reisner complex; tree polynomials Martin J.L.: The slopes determined by n points in the plane. Duke Math. J. 131(1), 119--165 (2006) Planar graphs; geometric and topological aspects of graph theory, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Configurations and arrangements of linear subspaces The slopes determined by \(n\) points in the plane | 0 |
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 d-dimensional regular local ring with maximal ideal \({\mathfrak m}\), and let \(\lambda\) be an element of \({\mathfrak m}\), \(\lambda\not\in {\mathfrak m}^ 2\). The question addressed in this paper is whether the ring \(A[\lambda^{-1}]\) is super-regular, in the sense that all its maximal ideals can be generated by d-1 elements. The answer is known to be positive for \(d\leq 2\) by unique factorization, and for \(d=3\) by a theorem of \textit{O. Gabber} [in Groupes de Brauer, Sémin., Les Plans-sur-Bex 1980, Lect. Notes Math. 844, 129-209 (1981; Zbl 0472.14013)]. The present paper gives a positive answer when A is a localization of an algebra finitely generated over a field, and A/\({\mathfrak m}\) is infinite. strongly regular rings; complete intersection; localization over a field; super-regular rings Budh Nashier, Strongly regular rings, J. Algebra 137 (1991), no. 1, 206 -- 213. Regular local rings, Linkage, complete intersections and determinantal ideals, Complete intersections Strongly regular rings | 0 |
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 scheme \(X\subseteq \mathbb{P}^n= \mathbb{P}^n_k\) of ``fat points'' is a 0-dimensional scheme defined by a homogeneous ideal
\[
I={\mathfrak p}_1^{m_1}\cap {\mathfrak p}_2^{m_2}\cap \dots\cap{\mathfrak p}_s^{m_s}\subseteq k[x_0,x_1,\dots, x_n],
\]
where \(k\) is an algebraically closed field with \(\text{char }k=0\) and each \({\mathfrak p}_i\) is the homogeneous ideal of a point \(P_i\). Every vector space \(I_t\) represents the linear system of hypersurfaces of degree \(t\) having multiplicity at least \(m_i\) at each \(P_i\), \(i=1,\dots, r\). Let the \(P_i\)'s be generic: it is an open problem (even for \(n=2\)), to determine the Hilbert function of \(X\). In this paper the case \(r\leq n+1\) is studied: under this hypothesis the ideal \(I\) is a monomial one and this allows to find the Hilbert function (here the ``genericity'' of the \(P_i\)'s is simply the fact that no \(s\) of them lie is an \((s-1)\)-linear space). Then the case \(r=2\) is considered, and a minimal system of generators and the graded Betti numbers of \(I\) are given by showing that in this case \(I\) is a splittable ideal, i.e. that there are two monomial ideals \(U,V\) such that \(I=U+V\), the minimal set of generators \(T(I)\) of \(I\) is the disjoint union of \(G(U)\), \(G(V)\) and there is a splitting function \(G(U\cap V)\to G(U)\times G(V)\) satisfying some technical hypotheses. The Betti numbers of \(I\) can be obtained from those of \(U, V\), which are computed. Eventually it is shown that \(I\) is splittable when \(m_1=\dots= m_r=2\). minimal resolution; fat points; Hilbert function; graded Betti numbers G. Fatabbi, Ideals of fat points and splittable ideals, inThe Curves Seminar at Queen's, Vol. 10, Queen's Papers in Pure and Applied Mathematics, Vol. 102, pp. 242--255. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Exposé II H: Ideals of fat points and splittable ideals | 0 |
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 begin by recalling the role that \textit{weak proregularity} of an ideal in a commutative ring has in \textit{derived completion} and \textit{adic flatness}. We also introduce the new concepts of \textit{idealistic} and \textit{sequential derived completion}, and prove a few results about them, including the fact that these two concepts agree iff the ideal is weakly proregular. Next we study the \textit{local nature of weak proregularity}, and its behavior w.r.t. \textit{ring quotients}. These results allow us to prove our main theorem, which states that \textit{weak proregularity occurs in the context of bounded prisms}. Prisms belong to the new groundbreaking theory of \textit{perfectoid rings}, developed by Scholze and his collaborators. Since perfectoid ring theory makes heavy use of derived completion and adic flatness, we anticipate that our results shall help simplify and improve some of the more technical aspects of this theory. derived completion; weak proregularity; adic flatness; prisms; perfectoid rings Perfectoid spaces and mixed characteristic, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Complete rings, completion, Derived categories, triangulated categories, Homological functors on modules of commutative rings (Tor, Ext, etc.), Resolutions; derived functors (category-theoretic aspects) Weak proregularity, derived completion, adic flatness, and prisms | 0 |
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 paper is devoted to the study of a classical and very much studied problem of interpolation in Algebraic Geometry (which revealed to be of interest also from many other points of view): given a linear system \({\mathcal L}_{n,d}(m_1,\dots,m_s)\) made by all hypersurfaces of degree \(d\) in \(\mathbb P^n\) having multiplicities \(m_1,\dots,m_s\) at \(s\) generic points \(p_1,\dots,p_s\), what is its dimension?
This problem is still open even for \(n=2\), and only partial results are known (with bounds on \(s\), on the multiplicities \(m_i\)'s or with other additional hypotheses). For \(n=2\) and \(n=3\), conjectures exist which describe the situation, and they state that the linear system will not have the expected dimension if it has a multiple base curve which is a \((-1)\)-curve for \(n=2\), while for \(n=3\) the multiple fixed locus could be either a line or a quadric surface through 9 of the points.
In the paper under review, the authors follow these ideas to determine which could be other ``critical'' base loci for higher values of \(n\). The case of linear spaces which meet the points with ``excessive'' multiplicity (with respect to \(d\)) is already known; the other cases studied here, for \(s=n+3\), are the rational normal cure passing through the points, its secant varieties and the cones over them.
In this way an interesting conjecture is given (Conjecture 6.4) which should cover all possible cases for \({\mathcal L}_{n,d}(m_1,\dots,m_{n+3})\) not to have the expected dimension. Evidence for the conjecture to hold is given in a series of cases.
Moreover, for \(\mathbb P^n\) blown up at \(n+3\) generic points, the cone of effective divisors and its facets are studied and competely described. interpolation; effective and movable cones; fat points; rational normal curves; secant varieties Brambilla, M.C., Dumitrescu, O., Postinghel, E: On the effective cone of \({\mathbb{P}}^n\) with \(n+3\) points. Exp. Math. \textbf{25}(4), 452-465 (2016) Divisors, linear systems, invertible sheaves, Hypersurfaces and algebraic geometry, Rational and ruled surfaces, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the effective cone of \(\mathbb{P}^n\) blown-up at \(n+3\) points | 0 |
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 0527.00002.]
Let \(k\subset K\) be a field extension of characteristic zero, (R,\({\mathfrak m})\) a valuation ring of K and \(S:=R\cap k\). Suppose that \({\mathfrak m}\cap S\) generates a prime ideal in R and \(\dim S<\infty.\) Then R is a filtered inductive limit of smooth S-algebras of finite presentation. When \(S=k\) and K is an algebraic function field over k then this result is a consequence of Zariski's uniformization theorem. - Let A be a noetherian ring and \(A^*\) the ultrapower of A with respect to a nonprincipal ultrafilter on \({\mathbb{N}}\). Then
(*) Is \(A^*\) a filtered inductive limit of smooth A-algebras of finite type?
If A is a DVR of characteristic zero then the answer is positive using the above result. Recently M. André noticed that (*) can have a positive answer only when A is excellent.- If (*) holds for an excellent regular local ring A then in particular \(A^*\) is a filtered inductive limit of regular local rings, i.e. \(A^*\) is a non-noetherian ring having some ''regular'' properties. It could be useful to study this class of rings which extend valuation rings of characteristic zero. field extension; valuation ring; ultrapower Popescu, D.: On Zariski's uniformization theorem. Lect. notes in math. 1056 (1984) Applications of logic to commutative algebra, Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.), Arithmetic theory of algebraic function fields, Valuation rings, Field extensions, Principal ideal rings, Relevant commutative algebra On Zariski's uniformization theorem | 0 |
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 henselian local ring essentially smooth over an infinite field \(k\). Assume that \(E=E^{**}\) is a contravariant bigraded functor on the category \(S\) of smooth schemes of finite type over \(k\), \(E:S\rightarrow \mathbf{Ab}\), that is representable in the stable \({\mathbb A}^1\)-homotopy theory and satisfies \( lE=0\) for \( l\in {\mathbb Z}\) invertible in \(R\). Assume that \(E\) is so called normalized with respect to the fraction field of \(R\). Let \(f:M\rightarrow \mathrm{Spec } R\) be a smooth affine morphism of (pure) relative dimension \(d\) and \(s_0,s_1:\mathrm{Spec } R \rightarrow M\) two sections of \(f\) such that \(s_0(P)=s_1(P)\), where \(P\) is the closed point of \(\mathrm{Spec } R\). Then the induced maps \(s_i^*:E(M)\rightarrow E(\mathrm{Spec }R)\), \(i=1,2\) are equal. This is a rigidity theorem, which extends the results of \textit{I. Panin} and the second author [J. Pure Appl. Algebra 172, No. 1, 49--77 (2002; Zbl 1056.14027)] obtained for an algebraically closed field \(k\) and it is similar to the rigidity theorem of \textit{O. Gabber} [Contemp. Math. 126, 59--70 (1992; Zbl 0791.19002)] obtained for algebraic \(K\)-theory. rigidity property; cohomology theory; stable \({\mathbb A}^1\)-homotopy category; Henselian rings Hornbostel, J.; Yagunov, S., Rigidity for Henselian local rings and \(\mathbb{A}^1\)-representable theories, Math. Z., 255, 2, 437-449, (2007), MR 2262740 Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups, \(K\)-theory and commutative rings, Henselian rings Rigidity for henselian local rings and \(\mathbb{A}^1\)-representable theories | 0 |
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 As an application of the results obtained in the first part of the paper [see the preceding review Zbl 0834.16032], the authors now show how the relative version of the strong second layer condition allows us to endow \(K(\lambda)\), the generically closed subset of \(\text{Spec}(R)\) canonically associated to a radical \(\lambda\) in \(R\)-mod, with structure sheaves, which generalize most previously studied sheaf constructions in noncommutative algebra.
First they develop some general machinery to construct structure sheaves through localization in a rather generic way. The sheaf construction itself, both in the symmetric and the classical case, depends upon the choice of a suitable topology on \(K(\lambda)\). In the classical case, they provide some extra background on localization at relative Ore sets, generalizing and strengthening the results obtained in the absolute, Noetherian case.
Then the results of the first part of the paper allow us to associate to any \(R\)-bimodule \(M\) both a symmetric and a classical sheaf of \(R\)- modules on \(K(\lambda)\), whose global sections reduce to the localization of \(M\) at \(\lambda\). This generalizes most previously studied symmetric and nonsymmetric sheaf constructions for modules over noncommutative rings. strong second layer condition; structure sheaves; sheaf constructions in noncommutative algebra; localizations; relative Ore sets; global sections DOI: 10.1007/BF00961454 Torsion theories; radicals on module categories (associative algebraic aspects), Ideals in associative algebras, Localization and associative Noetherian rings, Noncommutative algebraic geometry, Grothendieck categories, Localization of categories, calculus of fractions, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Relatively Noetherian rings, localization and sheaves. II: Structure sheaves | 0 |
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 moduli space \(M_0\) of semistable vector bundles of rank \(2\) with trivial determinant on a smooth curve of genus \(g\geq 3\) is a singular projective variety. \textit{C. S. Seshadri} constructed a desingularization \(S\) of \(M_0\) as the moduli space of parabolic vector bundles of rank \(4\), degree \(0\) with the endomorphism algebra of the underlying vector bundle isomorphic to a specialization of the matrix algebra \(M(2)\) [in: Proc. Int. symp. Algebraic Geometry, Kyoto 1977, 155--184 (1977; Zbl 0412.14005)]. Another desingularization of \(M_0\) was constructed by \textit{F. C. Kirwan} by successive blow ups of the singular strata [Ann. Math. (2) 122, 41--85 (1985; Zbl 0592.14011)]. The authors extend the isomorphism of stable points of \(K\) and \(S\) to a morphism \(K \to S\) which is a composite of two explicitly described blow downs. They prove that \(M_0\) has terminal singularities, trivial plurigenera and compute the stringy \(E\)-function of \(M_0\). They also compute the Betti numbers of Seshadri's desingularization in all degrees. The authors conjecture that the third desingularization of \(M_0\) by \textit{M. S. Narasimhan} and \textit{S. Ramanan} [in: C.P. Ramanujam - a tribute, Tata Inst. Fund. Res. Stud. Math., 8, 291--345 (1978; Zbl 0427.14002)] lies between \(K\) and \(S\) as the middle blow up. Kiem, Y.-H., Li, J.: Desingularizations of the moduli space of rank 2 bundles over a curve. Math. Ann. 330, 491--518 (2004) Vector bundles on curves and their moduli, Classical real and complex (co)homology in algebraic geometry, Motivic cohomology; motivic homotopy theory Desingularizations of the moduli space of rank 2 bundles over a curve | 0 |
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 prove the following result: Let \(X\) be a complex torus and \(M\) a normally generated line bundle on \(X\); then, for every \(p\geq 0\), the line bundle \(M^{p+1}\) satisfies Property \(N_p\) of Green-Lazarsfeld:
Let \(Y\) be a smooth complex projective variety of dimension \(n\) and let \(L\) be a very ample line bundle on \(Y\) defining an embedding \(Y\subset\mathbb{P}= \mathbb{P}(\text{H}^0(Y,L)^*)\); set \(S=S(L)= \text{Sym}^*\text{H}^0(L)\), the homogeneous coordinate ring of the projective space \(\mathbb{P}\), and consider the graded \(S\)-module \(G=G(L) =\bigoplus_d \text{H}^0(Y,L^d)\). Let \(E_*\) be a minimal graded free resolution of \(G\) (that is, an exact sequence with \(E_i\) free \(S\)-modules and such that the matrices of homogeneous polynomials giving the maps \(E_i\to E_{i=1}\) has no nonzero constant entries); the line bundle \(L\) satisfies property \(N_p\) \((p\in \mathbb{N})\) if and only if \(E_0=S\), \(E_i=\bigoplus S(-i-1)\) for \(1\leq i\leq p\). (Thus \(L\) satisfies property \(N_0\) if and only if \(Y\subset\mathbb{P} (\text{H}^0(L)^*)\) is projectively normal, that is, \(L\) is normally generated; \(L\) satisfies property \(N_1\) if and only if \(L\) satisfies property \(N_0\) and the homogeneous ideal \(I\) of \(Y\subset P(\text{H}^0(L)^*)\) is generated by quadrics; \(L\) satisfies property \(N_2\) if and only if \(L\) satisfies property \(N_1\) and the module of syzygies among quadratic generators \(Q_i\in I\) is spanned by relations of the form \(\sum L_iQ_i =0\), where \(L_i\) are linear polynomials; and so on.)
In 1984, \textit{Green} proved that if \(X\) is a Riemann surface of genus \(g\) and \(L\) is a holomorphic line bundle on \(X\) of degre \(2g+1+p\), then \(L\) satisfies property \(N_p\). Thus, if \(M\) is an ample line bundle on an elliptic curve, then \(M^{p+3}\) satisfies property \(N_p\) and Lazarsfeld formulated the following conjecture: If \(M\) is an ample line bundle on a complex torus, then, for every \(p\geq 0\), the line bundle \(M^{p+3}\) satisfies property \((N_p)\).
In the paper under review using the ideas of Kempf's paper [\textit{G. R. Kempf}, in: Algebraic analysis, geometry and number theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 225-235 (1989; Zbl 0785.14025)], we prove another theorem on syzygies of abelian varieties: If \(M\) is a normally generated line bundle on a complex torus \(X\), then, for every \(p\geq 0\), the line bundle \(M^{p+1}\) satisfies property \(N_p\). property \(N_p\); line bundle on complex torus Divisors, linear systems, invertible sheaves, Algebraic theory of abelian varieties, Syzygies, resolutions, complexes and commutative rings On syzygies of abelian varieties | 0 |
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 short note we discuss some aspects of what could be called algebraic spline geometry. We concentrate on the concept of generalized Stanley-Reisner rings, namely the rings \(C^r(\Delta)\) of piecewise polynomial \(r\)-smooth functions on a simplicial complex \(\Delta\) in \(\mathbb R^d\). It is well known that the geometric realization of the ordinary Stanley-Reisner ring (or face ring) \(C^0(\Delta)\) reflects the structure of the simplicial complex: the irreducible components, corresponding to the maximal faces, are linear and intersect each other transversally in the pattern of the simplicial complex. We believe that the geometrical realizations of the generalized Stanley-Reisner rings behave similarly, except that the irreducible components are no longer linear and that they intersect with the appropriate multiplicity. We formulate this as a conjecture, the local spline ring conjecture, and show that it indeed holds in two very simple cases. For more complex examples, where the results are still conjectural, see [\textit{N. Villamizar}, Algebraic geometry for splines. University of Oslo (PhD Thesis) (2012), Ch. 4] for the case \(d=2\), \(r=1\). algebraic spline geometry; Stanley-Reisner rings; simplicial complex; local spline ring conjecture Computer-aided design (modeling of curves and surfaces), Numerical computation using splines, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Computational aspects in algebraic geometry Algebraic spline geometry: some remarks | 0 |
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 the vector space \(V\) of \(r\times n\) matrices, acted upon by the group \(G=\mathrm{GL}_r(C) \times (C^{\times})^n\). Most points of \(V\) correspond to ordered \(n\)-tuples of points in the projective space \(P^{r-1}\). Such a tuple determines a matroid, a so-called representable matroid, via the dimensions of the spans of various subsets of the tuple. That matroid is invariant under the \(G\) action.
The paper under review is one of the many works that explore the relation between the matroid and certain aspects of the geometry of the orbit. In particular, consider the class of the orbit closure in the \(G\)-equivariant \(K\)-theory of the vector space \(V\). The authors conjecture (Conjecture 5.1) that the matroid determines this class.
Besides this conjecture the paper has two main results. In the first one the authors prove that certain coefficients of the \(K\)-class of the orbit closure are indeed determined by the matroid. In this result, the combinatorics of hook-shaped partitions/Schur functions play a role.
The other main result is the description of the ideal of the orbit closure using only the matroid -- up to radical. If the ideal was reduced this would prove the conjecture, and it indeed does prove it in special cases when the ideal is reduced. For example, rank 2 or corank 2 uniform matroids satisfy this property. matroid; determinantal variety; equivariant \(K\)-class Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Determinantal varieties, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Equivariant homology and cohomology in algebraic topology Matrix orbit closures | 0 |
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 \(R = K[x_1,\dots,x_n]/I\) be a zero-dimensional affine \(K\)-algebra, where \(K\) is an arbitrary field. Equivalently the authors consider a 0-dimensional affine scheme \(X \subset \mathbb{A}_K^n\) and consider the ideal \(I = I_X\) in \(K[x_1,\dots,x_n]\). It is of great interest to know if \(R\) is a complete intersection, i.e. if \(I\) can be generated by \(ht(I)\) elements. The authors produce effective algorithms for checking whether \(R\) is a complete intersection at a maximal ideal, whether \(R\) is locally a complete intersection, and whether \(R\) is a strict complete intersection. The main tool is a characterization given by
\textit{H. Wiebe} [Math. Ann. 179, 257--274 (1969; Zbl 0169.05701)]
of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. The authors work in the affine setting, but they point out that by using homogenization and dehomogenization we can move between the affine and the projective settings. Several algorithms have been given in the past, but this paper gives more general algorithms. An advantage of this approach is that it allows us to get a description of which generators of \(I\) form a regular sequence (respectively a strict regular sequence), and the approach works over an arbitrary base field \(K\). zero-dimensional affine algebra; zero-dimensional scheme; complete intersection; locally complete intersection; strict complete intersection; fitting ideal; border basis Linkage, complete intersections and determinantal ideals, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Computational aspects and applications of commutative rings, Complete intersections, Computational aspects in algebraic geometry Algorithms for checking zero-dimensional complete intersections | 0 |
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 purpose of this article is in some sense to illustrate the manner in which the classical methods of Galois descent work to impose constraints on the algebraic behaviour of local rings [for contrast, see \textit{A. Grothendieck}, ''Seminaire de géométrie algébrique'', Inst. Haut. Étud. Sci. (1962; Zbl 0159.504)]. By way of example we show in section \(2\) that a factorial local domain which arises as a normal extension (see the definition) of a regular local ring is necessarily a complete intersection. This is in sharp contrast to the situation for general domainns [e.g., see \textit{M.-J. Bertin}, C. R. Acad. Sci., Paris, Sér. A 264, 653-656 (1967; Zbl 0147.295)]; \textit{P. A. Griffith}, J. Pure Appl. Algebra 7, 303-315 (1976; Zbl 0338.13023)]; and \textit{R. M. Fossum} and \textit{P. Griffith}, Ann. Sci. École Norm. Supér., IV. Sér. 8, 189-200 (1975; Zbl 0303.13015)]''.
Definition of normal extension: ''Let R be a normal (i.e., integrally closed) domain having fraction field K. By a normal extension of R we mean a normal domain A which is a module finite extension of R such that the fraction field L of A is a Galois extension of K''. - Actually the author studies the following main problems for such extensions of regular local rings (all rings are commutative and noetherian): to be unramified in codimension one, to be unramified, A to be free as R-module and to be a complete intersection. And he obtains new important developments in difficult classical questions. In particular, the last result - ''a partial structure theory'' - is: ''Let A be a complete local factorial domain which is a normal extension of the regular local ring R. Let the Galois group G have order n and assume that R contains a primitive n-th root of unity and that n is a unit in R. Then A contains a normal domain D which is an abelian normal extension of R and is complete intersection of the form \(D\approx R[X_ 1,...,X_ T]/(X_ 1^{\ell_ 1}-p_ 1,...,X_ t^{\ell_ t}-p_ t)\). Moreover, A is unramified over D in codimension one.'' factorial local domain; normal extension; regular local ring; complete intersection Griffith P., Journal of Algebra 106 (2) (1987) Extension theory of commutative rings, Regular local rings, Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Complete intersections Normal extension of regular local rings | 0 |
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 Based on detail analysis of earlier works devoted to the study of resolutions of singularities in dimension 3 the author considers six quite formal axioms (openness, stability by normalization, etc.) characterizing certain regularity property in a given birational equivalence class of algebraic threefolds. Making use of the axioms and their combinations, he simplifies essentially proofs of some well-known results, obtains interesting applications concerning resolution of singularities of vector fields in dimension 3 [\textit{F. T. Cano}, Desingularization strategies for three-dimensional vector fields. Monografías de Matemática (Rio de Janeiro) 43. Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA). (1987; Zbl 1049.14006)] and a local version of Hironaka's strong factorization conjecture [\textit{H. Hironaka}, Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)] for birational morphisms of nonsingular projective threefolds in characteristic zero, he then adapts \textit{O. Zariski}'s proof of the patching theorem [Ann. Math. (2) 45, 472--542 (1944; Zbl 0063.08361)] to any regularity property satisfying these axioms, discusses some open problems and conjectures, etc. algebraic threefolds; resolution of singularities; valuations; local uniformization; regularity; singularities of vector fields; blow ups; nonsingular projective models; birational morphisms; patching theorem Piltant, O.: An axiomatic version of Zariski's pathching theorem. Universidad de Valladolid. Preprint (2008) Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Global theory of complex singularities; cohomological properties, Valuations and their generalizations for commutative rings, Singularities of vector fields, topological aspects An axiomatic version of Zariski's patching theorem | 0 |
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 \(P^ m(R)\) be a real projective space of dimension m and \(V^ r_{\alpha}\), \(\alpha =1,...,d\), \(d\leq m-r+1\), be a system of smooth submanifolds of dimension r, \(r\leq m-1\). This system is called algebraizable if in \(P^ m\) there exists a projective algebraic variety \(V^ r_ d\) of degree d and dimension r to which all \(V^ r_{\alpha}\) belong. For a complex projective space a condition for algebraizability is contained in Abel's well-known theorem [see \textit{P. A. Griffiths}, Invent. Math. 35, 321-390 (1976; Zbl 0339.14003)]. In \(P^ m(R)\) a local condition for algebraizability has been obtained in the following five cases:
i) \(r=m-1\) and \(d=3\) [the author, Sib. Mat. Zh. 14, 467-474 (1973; Zbl 0267.53005)];
ii) \(r=m-2\) and \(d=3\) [the author and the reviewer, Izv. Vyssh. Uchebn. Zaved. Mat. 1974, No.5(144), 12-24 (1974; Zbl 0297.53037)];
iii) \(r\geq 2\) and \(d=m-r+2\) [the reviewer, Itogi Nauki Tekh., Ser. Probl. Geom. 7, 173-195 (1975; Zbl 0547.53007); Tensor, New Ser. 36, 99-21 (1982; Zbl 0479.53014 and the author, Sib. Mat. Zh. 23, 6-15 (1982; Zbl 0505.53004)];
iv) \(r=m-1\) and \(d=4\) [the reviewer, Tensor, New Ser. 38, 179-197 (1982; Zbl 0513.53009)];
v) \(r=m-1\) and \(d\geq 2\) [\textit{J. A. Wood}, An algebraization theorem for local hypersurfaces in projective space, Ph. D. Thesis, Univ. of California, Berkeley, Calif., 1982 and \textit{J. A. Wood}, Duke Math. J. 51, 235-237 (1984)].
All these conditions for algebraizability constitute at the same time conditions for algebraizability of a d-web \(GW(d,m-r+1,r)\) of codimension r on the Grassmannian G(m-r,m). In the present paper the author gives in \(P^ m(R)\) a condition for algebraizability in the general case for any r satisfying the inequalities 2\(\leq r\leq m-1\) and \(d\leq m-r+1\). This condition is expressed as the vanishing of the sum of the asymptotic quadratic forms of \(d-m+r+1\) of the manifolds \(V^ r_{\alpha}\) with respect to a special hyperplane associated with the system of \(V^ r_{\alpha}\). real projective space; system of smooth submanifolds; algebraizability of a d-web; asymptotic quadratic forms M. A. Akivis, ''On a local condition for algebraizability of a system of subvarieties of a real projective space,'' Dokl. Akad. Nauk SSSR,272 (1983). Differential geometry of webs, Projective differential geometry, Varieties and morphisms, Questions of classical algebraic geometry, Projective techniques in algebraic geometry On a local condition for algebraizability of a system of submanifolds of real projective space | 0 |
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 formally equidimensional local ring \((R, \mathfrak{m})\), the multiplicity \(e(I)\) of an \(\mathfrak{m}\)-primary ideal \(I\) characterizes the integral closure of the ideal; that is, if \(J\) is another ideal with \(I \subseteq J\), then \(J\) and \(I\) have the same integral closure (or equivalently, \(I\) is a reduction of \(J\)) if and only if \(e(I)=e(J)\).
The \(j\)-multiplicity \(j(I)\) of the ideal \(I\), introduced by \textit{R. Achilles} and \textit{M. Manaresi} [J. Math. Kyoto Univ. 33, No. 4, 1029--1046 (1993; Zbl 0816.13019)], is an invariant defined for arbitrary ideals which coincides with the classical multiplicity \(e(I)\) in the \(\mathfrak{m}\)-primary case. Further expanding this concept, \textit{R. Achilles} and \textit{M. Manaresi} [Math. Ann. 309, No. 4, 573--591 (1997; Zbl 0894.14005)] defined an entire sequence of generalized multiplicities \(c_0(I), \ldots, c_d(I)\) with \(d=\dim R\) where the first element \(c_0(I)\) is the same as the \(j\)-multiplicity \(j(I)\).
\textit{H. Flenner} and \textit{M. Manaresi} [Math. Z. 238, No. 1, 205--214 (2001; Zbl 1037.13001)] proved that if \(I \subseteq J\) are arbitrary ideals in a formally equidimensional local ring, then \(I\) is a reduction of \(J\) if and only if \(j(I_\mathfrak{p})=j(J_\mathfrak{p})\) for all the prime ideals \(\mathfrak{p}\) in \(R\). This numerical characterization of the integral closure works for arbitrary ideals; however, it would be useful to have a characterization that only involves invariants that can be computed by considering only the ring \(R\) and not all of its localizations. From a computational point of view, this would be an essential requirement.
The sequence of generalized multiplicities \(c_0(I),\ldots, c_d(I)\) is the natural candidate for obtaining such a characterization. The reviewer [J. Pure Appl. Algebra 178, No. 1, 25--48 (2003; Zbl 1088.13501)] already proved that if \(I\subseteq J\) is a reduction, then \(c_k(I)=c_k(J)\) for all \(k=0,\ldots, d\). In this paper the authors show that the converse is true as well: if \((R, \mathfrak{m})\) is a formally equidimensional local ring and \(I \subseteq J\) are ideals such that \(c_k(I)=c_k(J)\) for all \(k\), then \(I\) is a reduction of \(J\). integral closure; reduction; multiplicity; \(j\)-multiplicity; multiplicity sequence Integral closure of commutative rings and ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Singularities in algebraic geometry Multiplicity sequence and integral dependence | 0 |
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 an excellent Henselian local ring of dimension \(d\). It is well known that \((R,m)\) has the so called strong Artin approximation property. Let \(f=(f_1,\ldots,f_r)\) be a system of polynomials of \(R[Y]\), \(Y=(Y_1,\ldots,Y_N)\) and \(\beta:{\mathbb N}\rightarrow {\mathbb N}\) the so called Artin function. The aim of this paper is to give conditions, when \(\beta\) can be chosen to be linear. If \(R\) is reduced then this happens when \(N=1\). If \(R\) is regular (resp. DVR) explicit calculations of linear Artin functions are included in the case when \((f)\) is a monomial (resp. determinantal) ideal. Let \(H_f\) be the ideal defining the smooth locus of \(R[Y]/(f)\) over \(R\) and \(t_1,\ldots, t_{d-1}\) a part of a system of parameters of \(R\). Using Elkik's ideas, here it shows that there exists a linear function \(\beta:{\mathbb N}\rightarrow {\mathbb N}\) such that for each \(n\in {\mathbb N}\) and each \(a\in R^N\) satisfying \(f(a)\subset m^{\beta(n)}\), and \(f(a)+H_f(a)\supset (t_1,\ldots,t_{d-1})\) there is \(b\in R^N\) such that \(f(b)=0\) and \(b\equiv a\) mod \(m^nR^N\). Artin approximation; excellent Henselian local ring; determinantal ideals; monomial ideals Dinh, T., On the linearity of Artin functions, J. Pure Appl. Algebra, 209, 2, 325-336, (2007) Étale and flat extensions; Henselization; Artin approximation, Excellent rings, Local deformation theory, Artin approximation, etc., Complete rings, completion On the linearity of Artin functions | 0 |
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 sixties, J. Nash initiated the study of the space of arcs of an algebraic variety \(X\), centered in some singular point, denoted by \(X_\infty^{\text{Sing}}\). This space has the structure of an (infinite dimensional) algebraic variety. J. Nash proved that \(X^{\text{Sing}}_\infty\) has a finite number of irreducible components, when the characteristic of the base field is zero. The proof of J. Nash is based on the existence of a resolution of singularities of \(X\). Therefore his result can be extended to any algebraic variety \(X\), for which there exists a resolution of singularities \(Y\longrightarrow X\).
The goal of this paper, is to strengthen this result. Roughly speaking, the authors show that if local uniformization holds for \(X\), then the space of arcs of \(X\), centred in some singular point, has a finite number of components. Their result is a bit more general, and is the following:
Theorem: Let \(X\) be a variety defined over a field \(k\), and \(Z\) be a subvariety of \(X\). Assume that for every \(z\in Z\), and every irreducible subvariety \(V\subset Z\), with \(z\in V\), local uniformization holds on \(V\) at \(z\). Then the space of arcs of \(X\) centered in a point of \(Z\), has a finite number of irreducible components. arc spaces; local uniformization Valuations and their generalizations for commutative rings, Local rings and semilocal rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Local uniformization and arc spaces | 0 |
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 0, \(S=k[x_1,\dots,x_s]\) and \(S_d\) the vector space of monomials of degree \(d\) in \(S\). The author investigates the generic initial subspace \(\text{gin} (V)\) of \(S_d\), \(V\) being a subspace of \(S_d\), with respect to the reverse lexicographic order. The main result is the following:
Put \(d=n+m\) and suppose \(\text{gin} (V)= W^n x_1^m\), where \(W=(x_1, \dots, x_r)\subseteq S_1\) and \(s\geq r\geq 3\). Then there exists a polynomial \(p\in S_m\) and a linear subspace \(W_n\subseteq S_n\) such that \(V=W_np\). The basics of the theory of generic initial spaces are provided. generic initial space; monomials; reverse lexicographic order Gunnar Fløystad, A property deducible from the generic initial ideal, J. Pure Appl. Algebra 136 (1999), no. 2, 127 -- 140. Polynomial rings and ideals; rings of integer-valued polynomials, Relevant commutative algebra A property deducible from the generic initial ideal | 0 |
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 part I, cf. ibid. 21, 75-179 (1985; Zbl 0573.17012).]
From the introduction: ``The purpose of the present paper is to introduce and to describe the flat structure on the ring of invariant \(\theta\)- functions for an extended affine root system.'' ``An extended affine root system \(R\) is a root system associated to a positive semi-definite Killing form with radical of rank 2. The extended Weyl group \(W_ R\) for \(R\) is an extension of a finite Weyl group \(W_ f\) by a Heisenberg group \(H_ R\). A Coxeter element \(c\) is defined in the group, whose power generates the center of the Heisenberg group.'' ``The \(W_ R\)-invariant ring \(S^ W\) is a ring of \(W_ f\)-invariant \(\theta\)-functions on a line bundle \(L\) over a family \(X\) of Abelian varieties, which is associate to \(H_ R\) and on which \(W_ f\) acts.''
``The flat structure on \(S^ W\) is roughly a certain particular system of homogeneous generators of the algebra \(S^ W\), whose linear span is uniquely characterized by admitting a \({\mathbb{C}}\)-inner product. The goal of the present paper is the construction of the flat structure for the root system with codimension 1. In the construction, the fact that the fixed point set of a Coxeter element is regular with respect to the Weyl group action plays an essential role.'' logarithmic connection; flat structure; ring of invariant \(\theta\)-functions; extended affine root system; Coxeter element A. Iqbal and C. Kozcaz, \textit{Refined Topological Strings and Toric Calabi-Yau Threefolds}, arXiv:1210.3016 [INSPIRE]. Infinite-dimensional Lie (super)algebras, Analytic theory of abelian varieties; abelian integrals and differentials, Infinite-dimensional Lie groups and their Lie algebras: general properties Extended affine root systems. II: Flat invariants | 0 |
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 an irreducible curve on a nonsingular projective surface \(S\) defined over an algebraically closed field of characteristic zero. Let \(\pi\:\widetilde S\to S\) be the minimal resolution of singularities of \(C\), i.e., a proper birational morphism from a nonsingular surface \(\widetilde S\), such that the proper transform \(\widetilde C\subseteq \widetilde S\) of \(C\) is smooth, and which does not strictly dominate other proper birational morphism with this property. A linear system \(\mathcal L\) on \(S\) is \textit{rational} if a general member of \(\mathcal L\) is an irreducible rational curve and \(\dim \mathcal L\geq 1\). The authors show (Theorem 2.8) that a rational linear system containing \(C\) exists if and only if \(C\) is rational and \((\widetilde C)^2\geq 0\). Furthermore, if such a system exists for \(C\) then it is contained in the linear system \(\pi_*|\widetilde C|\). rational curves; rational surfaces; linear systems; weighted cluster of singular points D. Daigle and A. Melle-Hernández: Linear systems of rational curves on rational surfaces , Mosc. Math. J. 12 (2012), 261-268. Divisors, linear systems, invertible sheaves, Rational and ruled surfaces Linear systems of rational curves on rational surfaces | 0 |
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 study the O-cycles in the projective spaces over an algebraically closed field of arbitrary characteristic in connection with the degree bound for defining equations (Castelnuovo bound) of irreducible nondegenerate singular curves. A finite set S in a projective space H is called in \textit{linear semi-uniform position} if S spans H and for any two linear subspaces \(L\), \(M\) of \(H\) with \(\dim(L)=\dim(M)\) and \(L\) (resp. \(M\)) spanned by \(L\cap S\) (resp. \(M\cap S)\) one has \(\text{card}(L\cap S)= \text{card}(M\cap S)\). If \(C\subset {\mathbb{P}}^ r\) is an irreducible non-degenerate curve and H a general hyperplane of \({\mathbb{P}}^ r\) then \(C\cap H\) is in linear semi-uniform position in H.
In the first part of the paper the main result is the theorem 0.1: for any finite subset \(S\) in \({\mathbb{P}}^ m\), \(S\) in linear semi-uniform position, and for any integer \(t>0\) one has \(h^ 0({\mathbb{P}}^ m,{\mathcal O}_{{\mathbb{P}}^ m}(t))-h^ 0({\mathbb{P}}^ m,{\mathcal I}_ S(t))\geq \min (\text{card}(S),mt+1)\). The bound given by the theorem can be improved if the set S is not in linear general position. The author makes attention to results exposed by \textit{J. Harris} and \textit{D. Eisenbud} in ``Curves in projective space'', Sémin. Math. Supér. 85 (1982; Zbl 0511.14014), chapter III, on Castelnuovo's theory [see also \textit{Ph. Griffith} and \textit{J. Harris}: ``Principles of algebraic geometry'' (1978; Zbl 0408.14001), chapter 2, {\S} 2, for the basic formula \(g\leq \pi (d,r)=({1\over 2})m(m-1)(n-1)+m\epsilon]\). With a review of the proof of this formulagiven by J. Harris and D. Eisenbud and using the above theorem (0.1) the author proves that the Castelnuovo bound on the arithmetic genus holds for singular projective curves in any characteristic.
In the second part of the paper the results of J. Harris and D. Eisenbud are generalised to singular curves on irreducible reduced and non- degenerate curves which extend the classical results of Castelnuovo. The method is to study attentively the very strange curves and to use the results on sets in linear semi-uniform position obtained as intersection with a general hyperplane. O-cycles; arbitrary characteristic; Castelnuovo bound; singular curves; linear semi-uniform position; arithmetic genus E. Ballico, On singular curves in the case of positive characteristic,Math. Nachr. 141 1989), 267--273. Algebraic cycles, Divisors, linear systems, invertible sheaves, Singularities of curves, local rings On singular curves in the case of positive characteristic | 0 |
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 How is the resolution of the ideal of a set of distinct generic points in \({\mathbb{P}}^n\) like? A conjecture about the graded Betti numbers of such resolutions (known as the ``minimal resolution conjecture'', MRC) was given by \textit{A. Lorenzini} [J. Algebra 156, No. 1, 5-35 (1993; Zbl 0811.13008)], and it has been proved in many cases and even asymptotically (when the number of points is much bigger then \(n\), by a result of Hirschowitz and Simpson).
Examples found computationally (Schreyer 1993) suggested nevertheless that the MRC could be false in general, even if no ``geometrical reason'' for those (three) counterexamples was known.
In the beautiful paper under review, such reason is found, enclosed in the theory of ``Gale transforms'' (they could be viewed as duality of linear series on a finite Gorenstein scheme), a way to associate to a set \(\Gamma \) of \(\gamma \geq r+3\) distinct points in \({\mathbb{P}}^r\) another set \(\Gamma ^\prime \) of \(\gamma \) points in \({\mathbb{P}}^s\), with \(s={\gamma -r -2}\).
The idea, expressed in classical language is this: With an appropriate choice of the coordinates, we can suppose that the coordinates of the points of \(\Gamma \) are the rows of a \((r+1)\times \gamma\) matrix \((I_{r+1}|B)\); then the coordinates of the points in \(\Gamma ^\prime \) are the rows of the matrix \((B^T|I_{s+1})\).
In the paper the relation among graded Betti numbers of \(\Gamma \) and of \(\Gamma ^\prime\) are found, and the ``mystery'' of the counterexamples to MRC is solved, moreover an infinite family of counterexample is determined (in any \({\mathbb{P}}^r\) with \(r\geq 6\), \(r\neq 9\)). Gale duality; generic points; minimal resolution conjecture Eisenbud, D.; Popescu, S., Gale duality and free resolutions of ideals of points, Invent. Math., 136, 419-449, (1999) Syzygies, resolutions, complexes and commutative rings, Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Gale duality and free resolutions of ideals of points | 0 |
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 variations of mixed Hodge structures (VMHS) associated with a pencil \(\mathcal{X}\) of equisingular hypersurfaces of degree \(d\) in \(\mathbb{P}^4\) with only ordinary double points as singularities, as well as the variations of Hodge structures (VHS) associated with the desingularization of this family \(\widetilde{\mathcal{X}}\). The notion of a set of singular points being in \textit{homologically good position} is introduced, and, by requiring that the subset of nodes in (algebraic) general position is also in homologically good position, we can extend Griffiths' description of the \(F^2\)-term of the Hodge filtration of the desingularization to this case, where we can also determine the possible limiting mixed Hodge structures (LMHS). The particular pencil \(\mathcal{X}\) of quintic hypersurfaces with 100 singular double points with 86 of them in (algebraic) general position that served as the starting point for this paper is treated with particular attention. variations of Hodge structures; Calabi-Yau manifolds; computational aspects in higher dimensional varieties; transcendental methods; Hodge theory Variation of Hodge structures (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Algebraic cycles, \(3\)-folds Variation of mixed Hodge structures associated to an equisingular one-dimensional family of Calabi-Yau threefolds | 0 |
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 fixed singularities imposed on members of a linear system of surfaces in \(\mathbb P^3_{\mathbb C}\) by its base locus \(Z\). For a 1-dimensional subscheme \(Z \subset \mathbb P^3\) with finitely many points \(p_i\) of embedding dimension three and \(d \gg 0\), we determine the nature of the singularities \(p_i \in S\) for general \(S \in |H^0 (\mathbb P^3, I_Z (d))|\) and give a method to compute the kernel of the restriction map from \(\mathrm{Cl}S \to \mathrm{Cl}\mathcal O_{S,p_{i}}\). One tool developed is an algorithm to identify the type of an \(\mathbf A_n\) singularity via its local equation. We illustrate the method for representative \(Z\) and use Noether-Lefschetz theory to compute \(\mathrm{Pic}S\). Brevik, J.; Nollet, S., Picard groups of normal surfaces, J. singul., 4, 154-170, (2012) Picard groups, Deformations of singularities, Families, moduli of curves (algebraic), Plane and space curves Picard groups of normal surfaces | 0 |
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 \(\mathcal A\) be an arrangement of \(n\) hyperplanes in \(\mathbb C^\ell\). Let \(\Bbbk\) be a field and \(A=\oplus_{p=0}^\ell A^p\) the Orlik-Solomon algebra of \(\mathcal A\) over \(\Bbbk\). The \(p^{\mathrm{th}}\) \textit{resonance variety} of \(\mathcal A\) over \(\Bbbk\) is the set \(\mathcal R^p(\mathcal A,\Bbbk)\) of one-forms \(a\in A^1\) annihilated by some \(b\in A^p\backslash(a)\). This notion arises naturally from consideration of cohomology of the rank-one local systems generated by single-valued branches of \(\mathcal A\)-master functions \(\Phi_a\).
For the most part we focus on the case \(p=1\). We will describe the features of \( \mathcal R^1(\mathcal A,\Bbbk)\) for \(\Bbbk=\mathbb C\) and also for fields of positive characteristic, and their connections with other phenomena. We derive simple necessary and sufficient conditions for an element \(a\) to lie in \(\mathcal R^1(\mathcal A,\Bbbk)\), and consequently obtain a precise description of \(\mathcal R^1(\mathcal A,\Bbbk)\) as a ruled variety. We sketch the description of components of \(\mathcal R^1(\mathcal A,\mathbb C)\) in terms of multinets, and the related Ceva-type pencils of plane curves. We present examples over fields of positive characteristic showing that the ruling may be quite nontrivial. In particular, \(\mathcal R^1(\mathcal A,\Bbbk)\) need not be a union of linear varieties, in contrast to the characteristic-zero case. We also give an example for which components of \(\mathcal R^1(\mathcal A,\Bbbk)\) do not intersect trivially.
We discuss the current state of the classification problem for Orlik-Solomon algebras, and the utility of resonance varieties in this context. Finally we sketch a relationship between one-forms \(a \in\mathcal R^p(\mathcal A,\Bbbk)\) and the critical loci of the corresponding master functions \(\Phi_a\). For \(p=1\) we obtain a precise connection using the associated multinet and Ceva-type pencil. arrangement of hyperplanes; Orlik-Solomon algebra; local system cohomology; resonance variety; master function Relations with arrangements of hyperplanes, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Homology with local coefficients, equivariant cohomology, Pencils, nets, webs in algebraic geometry Geometry and combinatorics of resonant weights | 0 |
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 regular local ring of dimension \(d \geq 2\) and \(O\) the closed point of \(X=\text{Spec} R\). A constellation \(C\) is a family \(\{Q_ 0,Q_ 1,\dots,Q_ n\}\) of infinitely near points to \(O\), with \(Q_ 0=O\), and \(Q_ i \in Z_ i\), where \(Z_ i\) is the variety obtained by the blowing up \(\sigma_ i:Z_ i \to Z_{i-1}\) of center \(Q_{i-1}\). For any point \(Q\) of \(C\) we denote by \(B_ Q\) the exceptional divisor of the blowing up of \(Q\) and we denote by \(E_ Q\) (resp. \(E^*_ Q)\) the strict (resp. total) transform of \(B_ Q\) in \(Z=Z_{n+1}\). -- We say that \(P\) is proximate to \(Q\), and we note \(P \to Q\), if \(P\) belongs to the strict transform of \(B_ Q\), then the proximity matrix is the \((n+1) \times(n+1)\) matrix \(M=(\mu_{ij})\) defined by \(\mu_{ii}=1\), \(\mu_{ij}=-1\) if \(Q_ i \to Q_ j\) and \(\mu_{ij}=0\) elsewhere. A cluster is a weighted constellation \(A=(C,\underline m)\) where \(\underline m=(m_ 0,\dots,m_ n) \in \mathbb{Z}^{n+1}\). To any cluster \(A\) we associate the divisor \(D(A)= \sum m_ iE^*_ i\), and we get the relation \(D(A)=\sum d_ iE_ i\) where \(\underline d=M^{-1} \underline m\). -- We say that an ideal \(I\) of \(R\) is finitely supported if there exists a constellation \(C\) such that \(I {\mathcal O}_ Z\) is invertible on \(Z\), where \(p:Z \to X\) is the map defined by \(C\). To the ideal \(I\) we associate the cluster \(A_ I=(C_ I,\underline m)\), where \(C_ I\) is the minimal constellation such that \(I {\mathcal O}_ Z\) is invertible and where \(m_ i\) is the multiplicity of the weak transform \(F_ i\) of \(I\) on the ring \({\mathcal O}_{Z_ i,Q_ i}\). The Cartier divisor \(D_ I=\sum d_ i E_ i\) associated to the invertible sheaf \(I {\mathcal O}_ Z\) is equal to \(\sum m_ iE^*_ i\). -- Then the authors get the following results:
The map \(I\) (complete finitely supported ideal) \(\mapsto A_ I\) (cluster) is injective.
For any constellation \(C\), the image of \(\{I\) complete finitely supported ideal \(/I {\mathcal O}_ Z\) invertible\} by \(I \mapsto D_ I\) is \(\{D\) exceptional divisor \(/p^*p_ * {\mathcal O}_ Z(-D) \to {\mathcal O}_ Z(- D)\) is onto\}.
The authors obtain also inequalities for the intersection numbers: \((- D_ I)^{d-k} \cdot E_{i_ 1} \dots E_{i_ k} \geq 0\), and they prove that \(p:Z \to X\) is an embedded desingularization of the hypersurface defined by a general element of \(I\).
Let \(X \simeq k^ d\) be a toric variety, we define a toric chain of infinitely near points \(Q_ i\) as a constellation \(C\) such that the morphisms \(\sigma_ i\) are toric. Then we can associate to \(C\) a family of integers \((a_ 1,\dots,a_ n)\) which determine the subdivisions of the cone. For these constellations the authors characterise the \(*\)- simple monomial ideals in the sense of Zariski-Lipman. constellation; infinitely near points; strict transform; proximity matrix; cluster; intersection; toric variety; toric chain of infinitely near points Campillo, A., Gonzalez-Sprinberg, G., Lejeune-Jalabert, M.: Amas, ideaux complets et chaines toriques. Comptes Rendus de l'Académie de Sciences - Série I - Mathématiques 315, 987-990 (1992) Infinitesimal methods in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Clusters, finitely supported ideals and toric chains | 0 |
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 \(\mathfrak{a}\) an ideal of \(R\). The multiplier ideals \(\mathcal{J}(R, \mathfrak{a}^t)\) in equicharacteristic 0 and test ideals \(\tau(R,\mathfrak{a}^t)\) in equicharacteristic \(p>0\) give a measure of the singularites of \(V(\mathfrak{a})\) scaled by nonnegative \(t\). The two main results of this paper are to give a mixed characteristic analog to these notions by introducing perfectoid test ideals \(\tau(R, \mathfrak{a}^t)\) and proving that the standard properties that hold for both multiplier ideals and test ideals such as:
\begin{itemize}
\item the containments between the multiplier/test ideals for comparable ideals scaled respect to a fixed exponent \(t\) and the reverse containments between the multiplier/test ideals for comparable scalings of a fixed ideal (Proposition 3.3),
\item the unambiguity of the exponents for positive integral powers (Proposition 3.7),
\item that the multiplier/test ideal is neither too small nor too big (Proposition 3.9 and Theorem 5.11), and
\item the subadditivity of multiplier/test ideals (Theorem 4.4)
\end{itemize}
also hold for perfectoid test ideals.
For a regular local ring \(R\) of mixed characteristic \((0,p)\), \(R[1/p]\) is equicharacteristic 0. The authors show that in this setting that the perfectoid test ideal \(\tau(R,\mathfrak{a}^t)\) extended to the ring \(R[1/p]\) is contained in the multiplier ideal of the ideal \((\mathfrak{a} R[1/p])^t\) in \(R[1/p]\) and conjecture the reverse containment also holds. They also define an asymptotic perfectoid test ideal of a graded sequence of ideals which they apply to the graded sequence the \(n\)th symbolic powers of a radical ideals \(I\) in a regular ring of height at most \(h\) to show \( I^{(hn)} \subseteq I^n\). The paper concludes with an illustration of the perfectoid test ideals of a nonnegatively scaled ideal generated by a single monomial in a mixed characteristic power series ring and a few open questions. symbolic power; multiplier ideal; test ideal; perfectoid algebra Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Multiplier ideals, Perfectoid spaces and mixed characteristic Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers | 0 |
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 \(d\)-dimensional Noetherian ring of characteristic \(p>0\). \(R\) is said to have FFRT (finite \(F\)-representation type) if there is a finite set of isomorphism classes of finitely generated indecomposable modules \(\{M_0, \ldots, M_n\}\) such that for any \(e \in \mathbb{N}\) there are \(c_{i,e} \geq 0\), such that \[R^{1/p^e} \cong M_0^{\oplus c_{0,e}}\oplus M_1^{\oplus c_{1,e}}\oplus \cdots \oplus M_n^{\oplus c_{n,e}}.\] The generalized \(F\)-signature of \(M_i\) with respect to \(R\) is \(s(M_i,R):=\underset{e \rightarrow \infty}\lim\displaystyle\frac{c_{i,e}}{p^{ed}}\).
A Hibi ring is a special type of toric ring defined via a poset. For toric rings \(R\) of characteristic \(p\), it is known that \(R\) has FFRT and the indecomposable modules of \(R\) are the conical divisors of \(R\). The goal of this nice paper is to determine the generalized \(F\)-signatures for the conical divisors of a Hibi ring.
The main theorem determines the generalized \(F\)-signature for a conical divisor of a Segre product of polynomial rings of dimension \(d\), which is a Hibi ring, in terms of the number of elements of the symmetric group on a set of \(d\) elements which certain descent properties. The authors claim that the methods used to prove this result can also be used to determine the generalized \(F\)-signature for a conical divisor for other Hibi rings; their running example of a Hibi ring which is not a Segre product provides an illustration of this claim. \(F\)-signature; strongly \(F\)-regular ring; finite \(F\)-representation type; toric ring; Hibi ring; conical divisor Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Combinatorial aspects of commutative algebra, Algebraic aspects of posets, Toric varieties, Newton polyhedra, Okounkov bodies, Commutative rings defined by binomial ideals, toric rings, etc. Generalized \(F\)-signatures of Hibi rings | 0 |
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 \(R=k[x_1,\ldots ,x_n]\) be a ring of polynomials over a field \(k\) of characteristic \(p>0\). There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules \(H^i_I(R)\) where \(I\subset R\) is an ideal. This algorithm has not been implemented because its complexity grows very rapidly with the growth of \(p\) which makes it impractical. In this paper we produce a modification of this algorithm that consumes a modest amount of memory. Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Integral closure of commutative rings and ideals, Multiplicity theory and related topics, Singularities in algebraic geometry Toward an efficient algorithm for deciding the vanishing of local cohomology modules in prime characteristic | 0 |
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 under review contains a significant new (negative) result about local monomialization of an inclusion of regular local rings along a valuation.
Recall that an inclusion of regular local rings \(R \to S\) of the same dimension \(n\), where \(S\) dominates \(R\) (i.e. \(m_S \cap R = m_R\)) is \textit{monomial} if there exist regular systems of parameters \(x_1, \ldots, x_n\) in \(R\), \(y_1, \ldots, y_n\) in \(S\), units \({\delta}_1, \ldots, {\delta}_n\) in \(S\) and a matrix \(A=(a_{ij})\) with nonnegative coefficients and nonzero determinant such that \(x_i={\delta}_i y_1^{a_{i1}} \cdots y_n^{a_{in}}, \quad 1 \leq i \leq n .\) Assume now that \(R\) and \(S\) above are excellent integral domains, with fields of fractions \(K\) and \(K^*\) respectively, where the induced inclusion \(K \to K^*\) is finite and separable, that \(\nu ^*\) is a valuation of \(K^*\) with valuation ring \(V_{\nu^*}\), \(S\) dominates \(R\) and \(\nu^*\) (i.e., \(V_{\nu ^*}\)) dominates \(S\). A \textit{weak local monomialization of \(R\to S\) along \(\nu ^*\)} is a commutative diagram of inclusions of regular local rings
\[
\begin{matrix} R_1&{\rightarrow}&S_1&{\subset}&{V_{\nu^*}}\\ {\uparrow}&&{\uparrow}&&\\ R&{\rightarrow}&S&& \end{matrix}
\]
where \(S_1\) dominate \(R_1\) and \(V_{\nu^*}\) dominates \(S_1\), such that the vertical arrows are birational and and \(R_1 \to S_1\) is binomial. If, in addition, the vertical arrows are compositions of monoidal transforms with regular centers, we talk about a \textit{local monomialization}.
In previous work the author proved that if \(K^*\) is a field of rational functions over a base field of characteristic zero, a local monomialization along a valuation always exists. In the present article he shows that this is not the case if the characteristic is positive. He proves that given any base field \(k\) of positive characteristic, containing at least three elements, there is a finite separable extension \(K \to K^*\) of \(n\) dimensional function fields and a valuation \(\nu^*\) of \(K^*\), regular local domains \(A\) and \(B\) whose fields of fractions are \(K\) and \(K^*\) respectively, where \(B\) dominates \(A\), a valuation \(\nu^*\) of \(K^*\) dominating \(B\), such that there is no weak monomialization of the inclusion \(A \to B\) along \(\nu ^*\).
The author proves this first for \(n =2\), by giving a specific example valid over any field with more than two elements. Actually, in this example \(K=k(u,v)\), \(K^*=k(x,y)\) (both fields of rational functions in two variables), where \(u=x^p(1+y)\), \(v=y^p+x\), and \(A=k[u,v]_{(u,v)}\) and \(B=k[x,y]_{(x,y)}\). To prove that there is no weak monomialization is complicated and ingenious, but the techniques involved are rather elementary. The transition from \(n=2\) to arbitrary dimension is achieved by ``adding more variables''. local ring; monomialization; valuation; positive characteristic DOI: 10.1007/s00208-014-1114-7 Global theory and resolution of singularities (algebro-geometric aspects), Birational geometry, Schemes and morphisms, Valuation rings, Excellent rings Counterexamples to local monomialization in positive characteristic | 0 |
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 Some results about the analytic branches of an algebraic affine variety along a singular subvariety are proved, using the theory of henselian rings. Precisely, let \(X=Spec(A)\), with A a noetherian domain, and Y a closed irreducible subvariety of X corresponding to the prime \({\mathfrak p}\) of A. The first result is that the global branches of X along Y, which by definition are the minimal primes of the henselization of the couple (A,\({\mathfrak p})\), correspond to the connected components of \(p^{-1}(Y)\), where \(p: X'\to X\) is the normalization morphism. Moreover, there is an open subset U of X such that there is a natural canonical correspondence between the global branches of U along \(U\cap Y\) and the branches of X at the generic point y of Y. A similar result is then proved for the geometric global branches of X along Y, i.e. the minimal primes of the strict henselization of the couple (A,\({\mathfrak p})\), replacing the branches of X in y with the geometric branches of X in y. Furthermore it is shown how to reconstruct the local rings of the branches at each point of a dense open subset of Y knowing the branches at the generic point y, under some conditions for the behaviour of X along Y. This result is finally extended to the general case, passing to a suitable étale covering of X.
For closely related results proved with completely different topological techniques see the paper by \textit{G. Tedeschi}, Boll. Unione Mat. Ital., VI. Ser., D, Algebra Geom. 4, No.1, 17-27 (1985; see the preceding review)]. analytic branches of an algebraic affine variety along a singular subvariety; henselian rings; geometric global branches Ramification problems in algebraic geometry, Henselian rings Global branches and parametrization | 0 |
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 survey the problems of resolution of singularities in positive characteristic and of local and global monomialization of algebraic mappings. We discuss the differences in resolution of singularities from characteristic zero and some of the difficulties. We outline Hironaka's proof of resolution for positive characteristic surfaces, and mention some recent results and open problems.
Monomialization is the process of transforming an algebraic mapping into a mapping that is essentially given by a monomial mapping by performing sequences of blow ups of nonsingular subvarieties above the target and domain. We discuss what is known about this problem and give some open problems. Cutkosky, Steven Dale, Ramification of valuations and counterexamples to local monomialization in positive characteristic, (2014), preprint Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Resolution of singularities in characteristic \(p\) and monomialization | 0 |
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 \(E_1, \ldots, E_k\) be a collection of linear series on an irreducible algebraic variety \(X\) over \(\mathbb{C}\) which is not assumed to be complete or affine. That is, \(E_i \subset H^0(X, \mathcal{L}_i)\) is a finite dimensional subspace of the space of regular sections of line bundles \(\mathcal{L}_i\). Such a collection is called overdetermined if the generic system \[s_1 = \ldots = s_k = 0,\] with \(s_i \in E_i\) does not have any roots on \(X\). In this paper we study consistent systems which are given by an overdetermined collection of linear series. Generalizing the notion of a resultant hypersurface we define a consistency variety \(R \subset \prod_{i = 1}^k E_i\) as the closure of the set of all systems which have at least one common root and study general properties of zero sets \(Z_{\mathbf{s}}\) of a generic consistent system \(\mathbf{s} \in R\). Then, in the case of equivariant linear series on spherical homogeneous spaces we provide a strategy for computing discrete invariants of such generic non-empty set \(Z_{\mathbf{s}}\). For equivariant linear series on the torus \((\mathbb{C}^\ast)^n\) this strategy provides explicit calculations and generalizes the theory of Newton polyhedra. resultants; Newton polyhedra; spherical varieties; complete intersections Solving polynomial systems; resultants, Toric varieties, Newton polyhedra, Okounkov bodies, Compactifications; symmetric and spherical varieties Overdetermined systems of equations on toric, spherical, and other algebraic varieties | 0 |
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 Das vorliegende Buch bietet neben einer systematischen Theorie der \({\mathfrak p}\)-adischen Zahlkörper auf elementarem Niveau und deren Standardanwendungen in der Theorie der algebraischen Zahlkörper eine Einführung in die Verwendung p-adischer Methoden zur Behandlung verschiedenartiger zahlentheoretischer Probleme (diophantische Gleichungen, Bernoulli-Zahlen, rekurrente Folgen, Potenzreihen algebraischer Funktionen), und es ist vor allem dieser letztere Aspekt, welcher den besonderen Reiz des Buches ausmacht. Vom Leser werden lediglich algebraische und zahlentheoretische Grundkenntnisse (wie sie in jeder ersten Einführungsvorlesung geboten werden) vorausgesetzt. Unter den zahlreichen Übungsaufgaben, welche jedes Kapitel des Buches beschließen, finden sich neben pädagogisch klug ausgewählten Standardbeispielen etliche auch für den Experten interessante zusätzliche Resultate und neue Beweisvarianten bekannter Sachverhalte.
Nun zum Inhalt der 13 Kapitel im einzelnen! Ch. 1, 2: Definition der Bewertung, einführende Beispiele, schwacher Approximationssatz, Komplettierung; als Anwendung der Satz von Staudt-Clausen und das Eisenstein-Kriterium für die Rationalität von Potenzreihen. Ch. 3: Archimedische Bewertungen, Satz von Ostrowski. Ch. 4: Nicht-archimedische Bewertungen: Henselsches Lemma, Nullstellen von Potenzreihen; Anwendungen auf rationale Matrizengruppen, diophantische Gleichungen (u.a. \(x^ 2- 17=2y^ 2\), \(x^ 2+7=2^ m)\) und spezielle Werte rekurrenter Folgen. Ch. 5: Einbettung endlich erzeugter Körper in \({\mathbb{Q}}_ p\) (Lech- Cassels); Satz von Mahler und Lech über rekurrente Folgen in Körpern der Charakteristik 0.
Ch. 6: Polynome über kompletten Körpern, Newton-Polygon, Weierstraßscher Vorbereitungssatz. Ch. 7, 8, 9: Klassische Theorie der Bewertungsfortsetzung im kompletten und im nicht-kompletten Fall; Beweis des ''lokalen'' Satzes von Kronecker-Weber und des quadratischen Reziprozitätsgesetzes; interessante Übungsaufgaben mit Jacobischen Summen. Ch. 10: Algebraische Zahlkörper; Idealtheorie, Endlichkeitssätze, ''Kronecker-Weber''; Anwendung auf diophantische Gleichungen (u.a. \(3x^ 3+4y^ 3+5z^ 3=0\), \(x^ 2+2=y^ 3\), \(x^ 3+dy^ 3=1)\). Ch. 11: Diophantische Gleichungen: Hasse-Prinzip für quadratische Formen über \({\mathbb{Q}}\), Kurven vom Geschlecht 0, ausführliche Diskussion von \(x^ 3+y^ 3+dz^ 3=0\) (Selmergruppe etc.). Ch. 12: Konstruktion p-adischer L-Funktionen mittels p-adischem Integral. Ch. 13: Beweis des \({\mathfrak p}\)-adischen Kriteriums von Borel und Dwork für die Rationalität von Potenzreihen. local fields; p-adic number fields; diophantine equations; Bernoulli numbers; recurrent series; power series of algebraic; functions; Weierstrass preparation theorem; Newton polygon; Kronecker-Weber theorem; Jacobi sums; Hasse principle; Selmer; group; p-adic L-functions; rationality of power series J. W. S. Cassels, \textit{Local fields}, London Mathematical Society Student Texts, Vol. 3, Cambridge University Press, Cambridge, 1986. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Algebraic number theory: local fields, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Algebraic number theory: global fields, Elementary number theory, Diophantine equations Local fields | 0 |
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 \(<\) be a monomial well-ordering on \(R=K[x_0, \dots,x_n]\), \(K\) an algebraically closed field For a monomial ideal \(I_0\) with Hilbert polynomial \(p(t)\) consider \(M_{I_0}= \{J\subset R\mid J\) homogeneous and saturated ideal and \(\text{in}(J)=I_0\}\). Here \(\text{in}(J)\) denotes the initial ideal of \(J\). It is proved that \({\mathcal M}_{I_0}\) carries the structure of a locally closed subscheme in \(\text{Hilb}^{p(t)}_{\mathbb{P}^n}\). \({\mathcal M}_{I_0}\) is affine if \(I_0\) is saturated. In this case, an explicit construction of the coordinate ring is possible. The set of all \({\mathcal M}_{I_0}\), \(I_0\) monomial with Hilbert polynomial \(p(t)\), leads to a natural stratification of \(\text{Hilb}_{\mathbb{P}^n}^{p(t)}\).
As an application, the singular locus of the component of \(\text{Hilb}^{4t+1}_{\mathbb{P}^4}\) containing the arithmetically Cohen-Macaulay curves of degree 4 is described. Hilbert scheme; stratification; Gröbner basis; Hilbert polynomial R. Notari and M.L. Spreafico, A stratification of Hilbert schemes by initial ideals and applications , Manuscr. Math. 101 (2000), 429-448. MR1759253 %(2001b:14008)
Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Computational aspects of higher-dimensional varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A stratification of Hilbert schemes by initial ideals and applications | 0 |
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 complex projective manifold of dimension \(n\) and \(L\) a positive line bundle (in this paper globally generated and big) on \(X\), \(K_X\) stands for the canonical bundle of \(X\). The study of adjoint linear systems of the type \(|K_X+kL|\) is of interest, specially when \(k\) is closed to \(n\). For example, when \(k=n-1\), the adjunction formula states that for \(C\) obtained as an intersection of \(n-1\) general elements of \(|L|\) the canonical bundle \(K_C\) is just the restriction of \(K_X+(n-1)L\) to \(C\). Then it is natural to think that properties of the morphism defined by \(|K_X+(n-1)L|\) are reflected in its behavior on \(C\), like birationality that perhaps is obstructed by the existence of hyperelliptic \(C\)'s of this type. The curves \(C\) are called curve sections of \(|L|\).
The paper under review provides necessary and sufficient conditions for the line bundles \(|K_X+kL|\) to be birational for \(k \geq n-1\) under the assumptions \(q(X)=0\) and non-emptyness of the linear system \(|K_X+(n-2)L|\). In fact, under these assumptions:
\(|K_X+(n+1)L|\) defines a birational map;
\(|K_X+nL|\) is not birational if and only if it defines a generic degree two map if and only if \(|L|\) provides a generic degree two morphism onto the projective space \(\mathbb{P}^n\);
\(|K_X+(n-1)L|\) is not birational if and only if the associated map has generic degree two if and only if there exist hyperelliptic curve sections of \(|L|\) if and only if all smooth irreducible curve sections of \(|L|\) are hyperelliptic if and only if \(|L|\) defines a generic degree two morphism onto a variety of minimal degree.
These results generalize and prove part of a conjecture by Gallego and Purnaprajna (see Conjecture 1.9 in [\textit{F. Gallego} and \textit{B. P. Purnaprajna}, Math. Ann. 312, No. 1, 133--149 (1998; Zbl 0956.14029)]). adjoint linear systems; hyperelliptic curve sections Divisors, linear systems, invertible sheaves, Rational and birational maps, Special divisors on curves (gonality, Brill-Noether theory), Calabi-Yau manifolds (algebro-geometric aspects) On the birationality of the adjunction mapping of projective varieties | 0 |
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 a set of vectors \(\mathcal{A}=\{{\mathbf a}_1,\ldots,{\mathbf a}_m\} \subset \mathbb{Z}^n\) and a Laurent polynomial ring \(K[t,t^{-1},s]:=K[t_1,t_1^{-1},\ldots,t_n,t_n^{-1},s]\) over a field \(K\), the associated homogeneous toric ideal \(I_{\mathcal{A}}\) is the kernel of the \(K\)-algebra homomorphism \(\phi:K[x_1,\ldots,x_n] \rightarrow K[\mathcal{A}]\) given by \(\phi(x_i)={\mathbf t}^{{\mathbf a}_i}s\) where \(K[\mathcal{A}]:=K[\{{\mathbf t}^{{\mathbf a}_i}s : {\mathbf a}_i \in \mathcal{A}\}]\) is the subalgebra of \(K[t,t^{-1},s]\) generated by all monomials \({\mathbf t}^{{\mathbf a}_i}s \). In the paper under review the author provide a geometric definition of combinatorial pure subrings. Given a subset \(\mathcal{A}'\) of \(\mathcal{A}\), the subring \(K[\mathcal{A}'] \subset K[\mathcal{A}]\) is a combinatorial pure subring of \(K[\mathcal{A}]\) if there exists a face \(F\) of the convex hull of \(\mathcal{A}\) such that \(\mathcal{A} \cap F=\mathcal{A}'\). Let \(\Phi \subset \mathbb{Z}^n\) be one of the classical root systems \({\mathbf B}_{n-1}\), \({\mathbf C}_{n-1}\), \({\mathbf D}_{n-1}\) and \({\mathbf BC}_{n}={\mathbf B}_{n} \cup {\mathbf C}_{n}\), \(n \geq 2\), and \(\Phi^{+}\) be the configuration consisting of all positive roots of \(\Phi\). The author use the geometric definition of combinatorial pure subrings together with the elimination property of the lexicographic order to explicitly construct a lexicographic squarefree initial ideal of \(I_{\Phi^{+}}\). Gröbner bases; toric ideals; convex polytopes; root systems H. Ohsugi, A geometric definition of combinatorial pure subrings and Gröbner bases of toric ideals of positive roots, Comm. Math. 56 (2007), 27--44. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Toric varieties, Newton polyhedra, Okounkov bodies, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes A geometric definition of combinatorial pure subrings and Gröbner bases of toric ideals of positive roots | 0 |
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 canonical ring \({S_{D} = \bigoplus_{d\geq0}H^{0}(X, \lfloor dD\rfloor)}\) of a divisor \(D\) on a curve \(X\) is a natural object of study; when \(D\) is a \({\mathbb{Q}}\)-divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. We study the generators and relations of \({S_D}\) for the simplest curve \({X = \mathbb{P}^1}\). When \(D\) contains at most two points, we give a complete description of \({S_D}\); for general \(D\), we give bounds on the generators and relations. We also show that the generators (for at most five points) and a Gröbner basis of relations between them (for at most four points) depend only on the coefficients in the divisor \(D\), not its points or the characteristic of the ground field; we conjecture that the minimal system of relations varies in a similar way. Although stated in terms of algebraic geometry, our results are proved by translating to the combinatorics of lattice points in simplices and cones. algebraic curve; canonical embedding; generators and relations; lattice point semigroups Evan O'Dorney, ''Canonical rings of \(\mathbb Q\) divisors on \(\mathbb P^1\)'', Annals of Combinatorics19 (2015) no. 4, p. 765-784 Special divisors on curves (gonality, Brill-Noether theory), Divisors, linear systems, invertible sheaves, Enumerative problems (combinatorial problems) in algebraic geometry Canonical rings of \(\mathbb{Q}\)-divisors on \(\mathbb{P}^1\) | 0 |
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 \((X,0)\) be a germ of an analytic space and \(f\) a germ of an analytic function on \(X\). The Milnor fibre \(F\) of \(f\) at zero is an analytic invariant of \(f\). Its topology can be reconstructed with the help of a desingularization \(P:Z\rightarrow X\) (in which the special fibre of \(f\) is a normal crossings divisor \(D\)) and a stratified projection of \(F\) over \(D\). These constructions give rise to Hironaka's filtration which is a valuative filtration on \(F\). Another technique due to the third author [in: Real and compl. singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 397--403 (1977; Zbl 0428.32008)] consists in considering a projection \(g\) of \(X\) on a disc and in using a polar curve (relative to \(f\) and associated to \(g\)) and a polar discriminant (or Cerf's diagram) to obtain a polar filtration on \(F\). In the present paper the authors show that these filtrations are homeomorphic and piecewise-diffeomorphic. This links invariants associated to the singularities of the projection to those associated to a desingularization. local Milnor fibre; polar construction; resolution of singularities Singularities in algebraic geometry, Local complex singularities, Milnor fibration; relations with knot theory Invariants of a desingularization and singularities of morphisms | 0 |
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 study linear systems on \((\mathbb{P}^{1})^{n}\). It is a classical question to ask about the dimension of a linear system of hypersurfaces in \(\mathbb{P}^{n}\) of a given degree passing through finitely many very general points with prescribed multiplicities. It is well-known that for \(n=2\) the Segre-Harbourne-Gimiliano-Hirschowitz conjecture predicts the dimension of such linear systems. It is natural to ask about possible generalizations of the SHGH conjecture for other values of \(n\) or another varieties.
Denote by \(\mathcal{L} = \mathcal{L}_{(d_{1}, \dots, d_{n})}(m_{1}, \dots, m_{r})\) the linear system of hypersurfaces in \((\mathbb{P}^{1})^{n}\) of degree \((d_{1}, \dots, d_{n})\) passing through \(r\) very general points \(q_{1}, \dots, q_{r}\) with multiplicities \(m_{1}, \dots, m_{r}\). \textit{The virtual dimension} of \(\mathcal{L}\) is
\[
\text{vdim}(\mathcal{L}) = \prod_{i=1}^{n}(d_{i}+1) - \sum_{i=1}^{r}{ n + m_{i}-1 \choose n} - 1.
\]
Then the expected dimension of \(\mathcal{L}\) is defined as \(\text{edim}(\mathcal{L}) = \max \{\text{vdim}(\mathcal{L}), -1 \}\). The inequality \(\text{dim}(\mathcal{L}) \geq \text{edim}(\mathcal{L})\) always holds. If \(\text{dim}(\mathcal{L}) > \text{edim}(\mathcal{L})\), then we say that \(\mathcal{L}\) is special.
For a given subset \(I \subset\{1, \dots, n\}\) we denote
\[
P_{I}: (\mathbb{P}^{1})^{n} \ni ([x_{1}:y_{1}], \dots, [x_{n}:y_{n}]) \mapsto ([x_{i}:y_{i}] \, : \, i \in I)\in (\mathbb{P}^{1})^{|I|}.
\]
Moreover, we denote by \(F_{j,I}\) the fiber of \(P_{I}\) through the point \(q_{j}\) for any \(j\). For a given vector \((d_{1}, \dots, d_{n}) \in \mathbb{Z}_{\geq 0}^{n}\) we denote by
\[
s_{I}:= \sum_{i \in I}d_{i} \text{ and } S_{I}:=1+ |I| +s_{I}
\]
with \(I \subset \{1, \dots,n\}\). By the assumption that points are very general one has \(F_{i,I}\cap F_{j,I} = \emptyset\) for \(i\neq j\). Then \textit{the fiber dimension} of \(\mathcal{L}\) is defined as
\[
\text{fdim}(\mathcal{L}) := \prod_{i=1}^{n}(d_{i}+1) - \sum_{1\leq j \leq r; \, I\subset \{1, \dots, n\}; \, S_{I} \leq m_{j}} (-1)^{|I|}{ m_{j}-S_{I}+n \choose n} -1.
\]
The fiber expected dimension is defined as \(\text{efdim}(\mathcal{L}) = \max \{\text{fdim}( \mathcal{L} ), -1\}\) and we say that \(\mathcal{L}\) is \textit{fiber special} if \(\dim(\mathcal{L}) > \text{efdim}(\mathcal{L})\). The first main result of the note is the following.
Theorem 1. For any linear system \(\mathcal{L}\) we have the following inequalities
\[
\dim(\mathcal{L}) \geq \text{efdim}(\mathcal{L}) \geq \text{edim}(\mathcal{L}).
\]
Another result gives us the following characterization of fiber non-special systems.
\textit{Theorem 2.} A linear system through two points in \((\mathbb{P}^{1})^{n}\) is fiber non-special.
Also if there are more than two points, then then there are examples of fiber special systems (see Example 5.2). linear systems; birational map; toric varieties Divisors, linear systems, invertible sheaves, Toric varieties, Newton polyhedra, Okounkov bodies Linear systems on the blow-up of \((\mathbb{P}^1)^n\) | 0 |
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 Hilbert scheme of \(m\) points \(\text{Hilb}_m(S)\) of a quasiprojective surface \(S\) parametrizes \(m\)-point subschemes of \(S\). The main purpose of this paper is to characterize the ring structure of the small equivariant quantum cohomology rings of \(\text{Hilb}_m(\mathcal{A}_n)\) for all \(m\) and \(n\), where \(\mathcal{A}_n\) is the crepant resolution of the \(A_n\) singularity \(\mathbb{C}/\mathbb{Z}_{n+1}\). This generalizes the work of Okounkov-Pandharipande for \(\text{Hilb}_m(\mathbb{C}^2)\).
The classical equivariant cohomology ring \(H^*_T(\mathcal{A}_n,\mathbb{Q})\) is generated by the \(n\) exceptional divisors \(\omega_i\) and the identity class \(1\). Moreover, \(H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\) has a basis, the Nakajima basis, labeled by cohomology weighted partitions \(\overrightarrow{\mu}=\{((\mu^{(1)},\gamma_{i_1}),\dots, (\mu^{(l)},\gamma_{i_l}))\}\) of \(m\), where \(\mu^{(1)}+\dots+\mu^{(l)}=m\) and \(\gamma_{i}\in H^*_T(\mathcal{A}_n,\mathbb{Q})\). Two divisors \(D:=-\{(2,1),(1,1),\dots,(1,1)\}\), a multiple of the boundary divisor of two point collisions, and \((1,\omega_i):=\{(1,\omega_i),(1,1),\dots,(1,1)\}\) play a central role in the paper.
The Fock space \(\mathcal{F}_{\mathcal{A}_n}\) modeled on \(\mathcal{A}_n\) is graded isomorphic to
\[
\mathcal{F}_{\mathcal{A}_n}=\bigoplus\limits_{m\geq0}H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\,,
\]
and the results of this paper are naturally stated in terms of this Fock space after extension of coefficients. The underlying reason is that \(\mathcal{F}_{\mathcal{A}_n}\otimes\mathbb{Q}(t_1,t_2)\) is isomorphic to a subspace in the basic representation of the affine Lie algebra \(\widehat{\mathfrak{gl}}(n+1)\). Quantum product on
\[
QH^*_T(\text{Hilb}_m(\mathcal{A}_n)):=H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q}) \otimes\mathbb{Q}(t_1,t_2)((q))[[s_1,\dots,s_n]]
\]
is a deformation of the classical cup product defined via three-point genus \(0\) Gromov-Witten invariants of the Hilbert scheme.
Two-point invariants can be encoded into an operator \(\Theta\) on the Fock space. To generate them, the authors define explicit representation-theoretic operators \(\Omega_0\), \(\Omega_+\) and prove that \(\Theta=(t_1+t_2)(\Omega_0+\Omega_+)\). This leads to their main result, explicit formulas for quantum multiplication operators \(M_D\) and \(M_{(1,\omega_i)}\) in terms of their classical counterparts \(M_D^{cl}\) and \(M_{(1,\omega_i)}^{cl}\):
\[
M_D=M_D^{cl}+(t_1+t_2)\,q\frac{\partial}{\partial q}(\Omega_0+\Omega_+)
\]
\[
M_{(1,\omega_i)}=M_{(1,\omega_i)}^{cl}+(t_1+t_2)\,s_i\frac{\partial}{\partial s_i}\Omega_+\,.
\]
One consequence of these formulas is that the multiplication operators are rational in \(q,s_1,\dots,s_n\) in contrast to the case of general surfaces, where only rationality in \(q\) is expected. They also imply a correspondence between multiplication by divisors in \(QH^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\) and in the equivariant Gromov-Witten theory of \(\mathcal{A}\times\mathbb{P}^1\) relative to the fibers at \(0\), \(1\) and \(\infty\). Under an additional conjecture that all joint eigenspaces of \(M_D\) and \(M_{(1,\omega_i)}\) are one-dimensional, and hence \(D\), \((1,\omega_i)\) generate \(QH^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\), the authors prove complete Gromov-Witten/Hilbert correspondence for \(\mathcal{A}_n\). If true, this is another special feature of these surfaces, which fails already for \(\mathbb{P}^2\), at least if no change of curve class variables is allowed. At the end of the paper the authors briefly discuss the quantum differential system \(q\frac{\partial}{\partial q}\psi=M_D\psi\), \(\,s_i\frac{\partial}{\partial s_i}\psi=M_{(1,\omega_i)}\psi\) and its monodromy, and indicate how their formulas can be extended to crepant resolutions of \(D\) and \(E\) singularities. Hilbert scheme of points; \(A_n\) singularity; small quantum cohomology; Nakajima basis; Gromov-Witten/Hilbert correspondence Maulik, D; Oblomkov, A, Quantum cohomology of the Hilbert scheme of points on \({\mathcal{A}}_n\)-resolutions, J. Am. Math. Soc., 22, 1055-1091, (2009) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of the Hilbert scheme of points on \(\mathcal {A}_n\)-resolutions | 0 |
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 Linear series are an important tool in the study of the geometry of rational maps from projective varieties \(X\). Nonempty linear series come from effective divisors on \(X\), while base-point-free linear series come from globally generated line bundles. In particular it is important to determine the effective and the base-point-free cones of divisors in the real Néron-Severi space \(N^1(X)\), and their closures the pseudo-effective \(\overline{\mathrm{Eff}}^1(X)\) and respectively nef \({\mathrm{Nef}}^1(X)\) cones. This problem is solved on projective bundles over curves by work of [\textit{Y. Miyaoka}, Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 449--476 (1987; Zbl 0648.14006)], and generalized for cycles of arbitrary codimension in [\textit{M. Fulger}, Math. Z. 269, No. 1--2, 449--459 (2011; Zbl 1230.14047)]. In particular from the work of Miyaoka it follows that if \(E\) is a vector bundle on \(C\), then \(\overline{\mathrm{Eff}}^1(\mathbb P(E))={\mathrm{Nef}}^1(\mathbb P(E))\) if, and only if, \(E\) is semi-stable. This is further equivalent to a certain normalized relative \(\mathcal O(1)\) divisor being nef on \(\mathbb P(E)\). The normalization is aiming to make \(\deg E=0\), in which case the condition is simply that \(E\) is nef. This last equivalence is generalized to arbitrary dimension in [\textit{I. Biswas} and \textit{U. Bruzzo}, Int. Math. Res. Not. 2008, Article ID rnn035, 28 p. (2008; Zbl 1183.14065)]. There it is proved that given a vector bundle \(E\) on a projective variety \(X\), then \(E\) is slope semi-stable and the discriminant \(\Delta(E)\) vanishes if, and only if, the analogous normalized class is nef.
This paper starts by investigating the first equivalence. Given \(X\) a projective variety and \(E\) a slope-semistable vector bundle with \(\Delta(E)=0\), the author's first result is that \(\overline{\mathrm{Eff}}^1(\mathbb P(E))={\mathrm{Nef}}^1(\mathbb P(E))\) if, and only if, \(\overline{\mathrm{Eff}}^1(X)={\mathrm{Nef}}^1(X)\). The proof relies on the above-mentioned result of [\textit{I. Biswas} and \textit{U. Bruzzo}, Int. Math. Res. Not. 2008, Article ID rnn035, 28 p. (2008; Zbl 1183.14065)]. The author also investigates the equivalence of nefness and pseudo-effectivity for cycles of codimension bounded above by the rank of \(E\), constructing examples on ruled surfaces and on surfaces of Picard rank 1. He also asks whether the converse holds, meaning if \(\overline{\mathrm{Eff}}^1(\mathbb P(E))={\mathrm{Nef}}^1(\mathbb P(E))\), does it follow that \(E\) is slope-semi-stable and \(\Delta(E)=0\)? However the tangent bundle to \(\mathbb P^n\) is a semi-stable counterexample with \(\Delta(T\mathbb P^n)\neq 0\) for all \(n\geq 2\).
This first result serves as a base example of a variety where weak Zariski decompositions for pseudo-effective divisors exist trivially. The author then proves that (possibly nontrivial) weak Zariski decompositions exist on fiber products of projective bundles over a curve. The proof uses an inductive argument based on a coning construction coming from [\textit{M. Fulger}, Math. Z. 269, No. 1--2, 449--459 (2011; Zbl 1230.14047)]. Nef cone; pseudo-effective cone; projective bundle; semistability; weak Zariski decomposition Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Embeddings in algebraic geometry, Projective techniques in algebraic geometry, \(n\)-folds (\(n>4\)), Minimal model program (Mori theory, extremal rays), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Pseudo-effective cones of projective bundles and weak Zariski decomposition | 0 |
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 study divisors on the blow-up of \(\mathbb{P}^n\) at points in general position that are non-special with respect to the notion of linear speciality. They describe the cohomology groups of their strict transforms via the blow-up of the space along their linear base locus and extend the result to non-effective divisors that sit in a small region outside the effective cone. As an application, they describe linear systems of divisors in \(\mathbb{P}^n\) blown-up at points in star configuration and their strict transforms via the blow-up of the linear base locus.
More in detail, denote by \(\mathcal{L}=\mathcal{L}_{n,d}(m_1,\dots, m_s)\) the linear system of hypersurfaces of degree \(d\) in \(\mathbb{P}^n\) passing through a collection of \(s\) points in general position with multiplicities at least \(m_1,\dots, m_s\geq 0\) respectively. The virtual dimension of \(\mathcal{L}\) is denoted by
\[
\mathrm{vdim}(\mathcal{L})={n+d \choose n}-\sum_{i=1}^s{n+m_i-1 \choose n}
\]
and its expected dimension as
\[
\mathrm{edim}(\mathcal{L})=\max\{\mathrm{vdim}(\mathcal{L}),0\}.
\]
If \(D\) is the strict transform of a general divisor in \(\mathcal{L}\) in the blow-up \(X\) of \(\mathbb{P}^n\) at the \(s\) points,
\[
D:=dH-\sum_{i=1}^sm_iE_i \tag{1.1}
\]
then \(\mathrm{vdim}(D):=\mathrm{vdim}(\mathcal{L})\) equals \(\chi(X,\mathcal{O}_X(D))\), the Euler characteristic of the sheaf on \(X\) associated with \(D\), while \(\dim(\mathcal{L})\) is the number of global section of \(\mathcal{O}_X(D)\), namely the dimension of the space \(H^0(X,\mathcal{O}_X(D))\).
We call \(\mathcal{L}\) (or \(D\)) nonspecial if its dimension equals the expected dimension, and special if \(\dim(\mathcal{L})>\mathrm{edim}(\mathcal{L})\). The speciality of \(\mathcal{L}\) (\(D\)) is defined to be the difference \(\dim(\mathcal{L}) -\mathrm{edim}(\mathcal{L})\).
In the last century the problem of computing the dimension of linear systems was studied with different techniques by many people. In the planar case, the Segre-Harbourne-Gimigliano-Hirschowitz conjectures predicts all special linear systems, while, in the case of \(\mathbb{P}^3\), there is an analogous conjectural classification of special linear systems formulated by Laface and Ugaglia.
Due to its complexity and mysterious geometry, the simple question of predicting and computing dimensions of such vector spaces is not even conjectured when \(n\) is four or higher. In the case of \(\mathbb{P}^n\) general results are rare and few things are known. For multiplicities higher than 2, the only general result known so far is a complete cohomological classification of the speciality of only linearly obstructed effective divisors, proved by \textit{M. C. Brambilla} et al. [Trans. Am. Math. Soc. 367, No. 8, 5447--5473 (2015; Zbl 1331.14007)]. One of the goal of this paper is to extend such a classification to the non-effective case.
In order to classify the special divisors, one has to understand first what are the obstructions, that is what are the varieties that whenever contained with multiplicity in the base locus of a given divisor force \(\mathcal{L}\) to be special.
The authors, in the direction of extending the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture to \(\mathbb{P}^n\) and possibly to other rational projective varieties, pose the following
Question 1.1. Consider any non-empty linear system \(\mathcal{L}\) in \(\mathbb{P}^n\). Let \(\tilde{X}\) be the smooth composition of blow-ups of \(\mathbb{P}^n\) along the (strict transforms of the) cycles of the base locus of \(\mathcal{L}\), ordered in increasing dimension. We denote by \(D\) a general divisor of the linear system \(\mathcal{L}\), and by \(\tilde{\mathcal{D}}\) the strict transform of \(D\) in \(\tilde{X}\). Does \(h^i(\tilde{X},\mathcal{O}_{\tilde X}(\tilde{\mathcal{D}}))\) vanish for all \(i\geq 1\)?
An affirmative answer to this question implies that \(\dim(\mathcal{L}) = \chi(\tilde{X},\mathcal{O}_{\tilde X}(\tilde{\mathcal{D}}))\), translating the classical dimensionality problem for linear systems into a Riemann-Roch formula for divisors living in subsequently blown-up spaces.
Denote now by \(\tilde{D}\) the strict transform of \(D\) in \(\tilde{X}\), the blow-up of \(X\) along the linear cycles of the base locus of \(D\). Clearly \(\tilde{D}\) is different from \(\tilde{\mathcal{D}}\) of Question 1.1 since the second one denotes the strict transform of \(D\) in \(\tilde{X}\), the blow-up of \(X\) along all -- linear and non linear -- cycles of the base locus of \(D\).
In this paper the authors use the concept of linear exprcted dimension instead of the one of expected dimension.
Definition 1.2. Given a linear system \(\mathcal{L}=\mathcal{L}_{n,d}(m_1,\dots, m_s)\), for any integer \(-1 \leq r \leq s - 1\) and for any multi-index \(I(r) = \{i_1,\dots,i_{r+1}\}\subseteq \{1,\dots, s\}\), define the integer
\[
k_{I(r)}:=\max(m_{i_1}+\cdots +m_{i_{r+1}}-rd,0).
\]
The linear virtual dimension of \(\mathcal{L}\) (or of \(D\)), denoted by \(\mathrm{ldim}(\mathcal{L})\), is the number
\[
\sum_{r=-1}^{s-1}\sum_{I(r)\subseteq\{1,\dots, s\}} (-1)^{r+1} \binom{n+k_{I(r)}-r-1}{n},
\]
where we set \(I(-1)=\emptyset\). The linear expected dimension of \(\mathcal{L}\) is defined as follows: it is 0 if \(\mathcal{L}\prec_s\mathcal{L}'\) and \(\mathrm{ldim}(\mathcal{L}') \leq 0\), otherwise it is the maximum between \(\mathrm{ldim}(\mathcal{L})\) and \(0\). Here we write \(\mathcal{L}\prec_s\mathcal{L}'\), for two linear systems \(\mathcal{L}_{n,d}(m_1,\dots,m_s)\) and \(\mathcal{L}_{n,d}(m'_1,\dots,m'_s)\) with the same degree, if \(m_i \geq m'_i\) for all \(i \in \{1,\dots,s\}\).
It is important to notice that this notion is well-defined not only for all effective divisors but also for non-effective ones. In this light, asking whether the dimension of a given linear system equals its linear expected dimension can be thought of as a refinement of the classical question of asking whether the dimension equals the expected dimension. If the answer to this question is affirmative, then \(\mathcal{L}\) is said to be a only linearly obstructed. Obviously, non-special linear systems are always only linearly obstructed. There exist linear systems that are linearly obstructed without being only linearly obstructed. For instance \(\mathcal{L}_{4,10}(6^7)\) contains all lines \(L_{ij}\), \(i,j \in \{1, \dots, 7\}\) with multiplicity two in its base locus as well as the rational normal curve through the seven points.
In [loc. cit.] it is proved that
Theorem 1.3. All non-empty linear systems of the form \(\mathcal{L}=\mathcal{L}_{n,d}(m_1,\dots, m_s)\) with \(s \leq n+2\) base points are only linearly obstructed. Moreover, if \(s \geq n + 3\) and
\[
\sum_{i=1}^s m_i \leq nd+\min(n-s(d),s-n-2), \quad 1\leq m_i \leq d,
\]
where \(s(d) \geq 0\) is the number of points of multiplicity \(d\), then \(\mathcal{L}\) is non-empty and only linearly obstructed.
For every effective divisor \(D\), let \(D_{(r)}\) denote the strict transform of \(D\) in the space \(X^n_{(r)}\) obtained as a sequence of blow-ups of \(\mathbb{P}^n\) along the linear base locus of \(D\) up to dimension \(r\), with \(r \leq n - 1\):
\[
D_{(r)}:=D- \sum_{\rho=1}^{r}\sum_{I(\rho)\subseteq\{1,\dots, s\}} k_{I(\rho)}E_{I(\rho)}.
\]
where \(E_{I(\rho)}\) denotes the (strict transform of the) exceptional divisor of the linear subspace of \(\mathbb{P}^n\) of dimension \(\rho\) spanned by the points parametrised by the multi-index \(I(\rho)\) and \(k_{I(\rho)}\) is the multiplicity with which the aforementioned subspace is contained in the base locus. Set \(\tilde{D}:=D_{(\overline{r})}\), where \(\overline{r}\) is the maximum dimension of the linear base locus.
In [loc. cit.] it is also proved that
Theorem 1.4. Given integers \(d\), \(m_1, \dots,m_s\), if \(s \leq n + 2\) and \(D\) is effective, the following statements hold.
\((a)\) \(h^0(D)=\mathrm{ldim}(D)\) and \(h^i(\tilde{D})=0\) for every \(i\geq 1\).
\((b)\) For any \(0\leq r\leq n-1\), \(h^i(D_{(r)})=0\) for every \(i\geq 1\) and \(i\not=r+1\), while
\[
h^{r+1}(D_{(r)})=\sum_{\rho=r+1}^{s-1}\sum_{I(r)\subseteq\{1,\dots, s\}} (-1)^{\rho -r-1} \binom{n+k_{I(\rho)}-\rho-1}{n}.
\]
One of the goal of this paper is to show that the same type of results as in Theorem 1.4 holds for larger classes of divisors, such as effective divisors with arbitrary number of general base points and non-effective divisors.
The authors also extend the formula in Theorem 1.4 to the case of any effective divisor, not necessarily only linearly obstructed, interpolating an arbitrary collection of general multiple points.
Theorem 1.5. Given integers \(d\), \(m_1, \dots,m_s\), consider the divisor \(D\) in (1.1). if \(D\) is effective, then for any \(0\leq r\leq n-1\) we have
\[
h^{r+1}(D_{(r)})=\sum_{\rho=r+1}^{s-1}\sum_{I(r)\subseteq\{1,\dots, s\}} (-1)^{\rho -r-1} \binom{n+k_{I(\rho)}-\rho-1}{n}+\sum_{\rho=r+1}^n(-1)^{\rho -r-1}h^\rho(\tilde{D}).
\]
In particular,
\[
h^0(D)=\mathrm{ldim}(D)+\sum_{\rho=r+1}^n(-1)^{\rho -r-1}h^\rho(\tilde{D}).
\]
Moreover, if \(h^i(\tilde{D})=0\), for all \(i\geq1\), then \(h^i(D_{(r)})=0\) for all \(i\not= r+1\).
The geometric interpretation is that for any effective divisor \(D\), every \(\rho\)-dimensional linear cycle \(L_{I(\rho)}\), for which \(k_{I(\rho)}\geq1\) and \(\rho\geq r+1\), gives a contribution with sign, \((-1)^{r-\rho+1}\), equal to
\[
\binom{n+k_{I(\rho)}-\rho-1}{n}
\]
to \(h^{r+1}(D_{(r)})\) and to the formula for \(\mathrm{ldim}(D)\). Moreover, such a contribution is zero when \(k_{I(\rho)}\leq \rho\).
The main result of this paper is a complete cohomological description of \(D_{(r)}\) in the following cases, where \(b:=b(D)=\sum_{i=1}^sm_i-nd\).
Theorem 1.6. Fix \(d\), \(m_1, \dots,m_s\). Statements \((a)\) and \((b)\) of Theorem 1.4 hold for all divisors \(D\) of the form (1.1) with \(m_i \leq d+1\) under the following hypothesis: \(s\leq n+1\) and \(b\leq n\), or \(s=n+2\) and \(b\leq 1\), or \(s\geq n+3\) and \(b\leq \min(n-s(d),s-n-2)\).
Moreover, if \(s\leq n+1\) then \(h^i(\tilde{D})=0\), for all \(i\leq n-1\), and \(h^n(\tilde{D})=\binom{b-1}{n}\) for \(b\geq n+1\) and zero otherwise. special systems; obstructions Dumitrescu, O., Postinghel, E.: Vanishing theorems for linearly obstructed divisors, arXiv:1403.6852 (2014) Divisors, linear systems, invertible sheaves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Hypersurfaces and algebraic geometry, Projective techniques in algebraic geometry Vanishing theorems for linearly obstructed divisors | 0 |
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