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\section{The four vertex theorem}
The classic 4-vertex theorem states that {\it the curvature of a smooth closed convex planar curve has at least four critical points}, see Figure \ref{ellipse} for an illustration.
\begin{figure}[hbtp]
\centering
\includegraphics[height=2in]{ellipse}
\caption{An ellipse and its evolute, the envelope of its normals or, equivalently, the locus of the centers of curvature. The cusps of the evolute correspond to the vertices of the curve.}
\label{ellipse}
\end{figure}
Since its discovery by S. Mukhopadhyaya in 1909, this theorem has generated a large literature, comprising various generalizations and variants of this result; see \cite{DGPV,GTT,OT3,Pak} for a sampler.
One approach to the proof of the 4-vertex theorem is based on the following observation: if a $2\pi$-periodic function $f(x)$ is $L^2$-orthogonal to the first harmonics, that is, to the functions $1, \sin x, \cos x$, then $f(x)$ must have at least four sign changes over the period.
The proof is simple: since $\int_0^{2\pi} f(x) dx =0$, the function $f(x)$ must change sign. If there are only two sign changes, one can find a linear combination $g(x)=c+a\cos x + b \sin x$ that changes sign at the same points as $f(x)$. Since the first harmonic $g(x)$ cannot have more than two sign changes, $f(x) g(x)$ has a constant sign, and $\int_0^{2\pi} f(x) g(x) dx \neq 0$, a contradiction. Discrete versions of this argument are in the hearts of our proofs presented below.
(In the 4-vertex theorem, one takes $f(x)=p'(x)+p'''(x)$, where $p(x)$ is the support function of the curve; then $p(x)+p''(x)$ is the curvature radius.)
The above observation is a particular case of the Sturm-Hurwitz theorem: {\it the number of zeros of a periodic function is not less than the number of zeros of its first non-trivial harmonic}, see \cite{OT3} for five proofs and applications of this remarkable result.
\begin{figure}[hbtp]
\centering
\includegraphics[height=2in]{Mukhopadhyaya}\quad
\includegraphics[height=2in]{Sturm}\quad
\includegraphics[height=2in]{Hurwitz}
\caption{Syamadas Mukhopadhyaya, Jacques Charles Fran\c{c}ois Sturm, and Adolf Hurwitz.}
\label{MSH}
\end{figure}
\section{Frieze patterns}
A frieze pattern is an array of numbers consisting of finitely many bi-infinite rows; each next row is offset half-step from the previous one. The top two rows consist of 0s and of 1s, respectively, the bottom two rows are the row of 1s and 0s as well, and every elementary diamond
$$
\begin{matrix}
&N&
\\
W&&E
\\
&S&
\end{matrix}
$$
satisfies the unimodular relation $EW-NS=1$. The number of non-trivial rows is called the width of a frieze pattern. Denote the width by $w$ and set $n=w+3$.
For example, a general frieze pattern with $w=2, n=5$ looks like this:
$$
\begin{array}{ccccccccccc}
\cdots&&1&& 1&&1&&\cdots
\\[4pt]
&x_1&&\frac{x_2+1}{x_1}&&\frac{x_1+1}{x_2}&&x_2&&
\\[4pt]
\cdots&&x_2&&\frac{x_1+x_2+1}{x_1x_2}&&x_1&&\cdots
\\[4pt]
&1&&1&&1&&1&&
\end{array}
$$
where the rows of 0s are omitted. These formulas appeared in the paper by Gauss ``Pentagramma Mirificum", published posthumously; Gauss calculated geometric quantities characterizing spherical self-polar pentagons, see Figure \ref{miri}. See also A. Cayley's paper \cite{Cay}. (According to Coxeter \cite{Cox} -- the very paper where frieze patterns were introduced -- the story goes further back, to N. Torporley, who in 1602 investigated the five ``parts" of a right-angled spherical triangle, anticipating by a dozen years the rule of J. Napier in spherical trigonometry.)
\begin{figure}[hbtp]
\centering
\includegraphics[height=2.5in]{mirificum1}
\caption{Pentagramma mirificum of Carl Friedrich Gauss.}
\label{miri}
\end{figure}
And here is a frieze pattern of width four whose entries are natural numbers:
$$
\begin{array}{ccccccccccccccccccccccc}
&&1&&1&&1&&1&&1&&1&&1&&1
\\[4pt]
&1&&3&&2&&2&&1&&4&&2&&1&
\\[4pt]
&&2&&5&&3&&1&&3&&7&&1&&2&
\\[4pt]
&1&&3&&7&&1&&2&&5&&3&&1&
\\[4pt]
&&1&&4&&2&&1&&3&&2&&2&&1&
\\[4pt]
&1&&1&&1&&1&&1&&1&&1&&1
\end{array}
$$
The very existence of such frieze patterns is surprising: the unimodular rule $EW-NS=1$ does not agree easily with the property of being a positive integer!
The frieze patters consisting of positive integers were classified by Conway and Coxeter \cite{CoCo}: they are in 1-1 correspondence with the triangulations of a convex $n$-gons by diagonals, and there are
$\frac{(2(w+1))!}{(w+1)!(w+2)!}$ (Catalan number) of them; see \cite{Hen} for an exposition of this beautiful theorem. For example, the above frieze pattern corresponds to the triangulation in Figure \ref{heptagon}.
\begin{figure}[hbtp]
\centering
\includegraphics[height=2in]{heptagon}
\caption{A triangulation of a heptagon: the labels are the number of the triangles adjacent to each vertex. These numbers comprise the first row of the frieze pattern.}
\label{heptagon}
\end{figure}
For a while, frieze patterns remained a relatively esoteric subject, but recently they have attracted much attention due of their significance in algebraic combinatorics and the theory of cluster algebras. I recommend a comprehensive contemporary survey of this subject \cite{Mor}.
\begin{figure}[hbtp]
\centering
\includegraphics[height=1.5in]{Conway}\quad
\includegraphics[height=1.5in]{Coxeter}
\caption{John Horton Conway and Harold Scott MacDonald Coxeter.}
\label{ConCox}
\end{figure}
Let us summarize the basic properties of frieze patterns relevant to this article. Denote by $a_i$ the entries of the first non-trivial row.
\begin{enumerate}
\item The NE diagonals of a frieze pattern satisfy the 2nd order linear recurrence (discrete Hill's equation)
$$V_{i+1} = a_i V_i - V_{i-1}$$
with $n$-periodic coefficients whose all solutions are antiperiodic, i.e., $V_{i+n}=-V_i$ for all $i$:
$$
\begin{array}{ccccccccccc}
0\ \ &&0&&0&&0\\
&1&&1&&1\\
&&a_1&&a_{2}&&a_{3}\\
&&&a_1a_2-1&&a_2a_3-1\\
&&\cdots&&a_1a_2a_3-a_1-a_3&&\cdots\\
\end{array}
$$
\item The solutions of the discrete Hill's equation can be thought of as polygonal lines $\ldots,V_1, V_2, \ldots$ in ${\mathbb R}^2$, with $\det (V_i,V_{i+1})=1$ and $V_{i+n}=-V_i$. Such polygonal line is well defined up to $\operatorname{SL}(2,{\mathbb R})$-action. The projections of the vectors $V_i$ to ${\mathbb {RP}}^1$ form an $n$-gon therein, well-defined up to a M\"obius transformation. For odd $n$, this correspondence between frieze patterns of width $n-3$ and projective equivalence classes of $n$-gons in the projective line is 1-1.
\item Label the entries as follows:
$$
\begin{array}{ccc}
&v_{i,j}&\\
v_{i,j-1}&&v_{i+1,j}\\
&v_{i+1,j-1}&
\end{array}
$$
with $a_i=v_{i-1,i+1}$.
Then one has $v_{i,j}=\det(V_i,V_j)$, explaining the glide reflection symmetry of the entries: $v_{i,j} = v_{j,i+n}$
The Conway-Coxeter article \cite{CoCo} starts with a description of the seven ornamental frieze patterns where the glide reflection symmetry is represented by
$
{\bf \large
\ldots b\quad p\quad b\quad p\quad b\quad p\ldots
}
$
and described as ``the relation between successive footprints when one walks along a straight path covered with snow". In Conway's nomenclature, this ornamental frieze pattern is called ``step", see Figure \ref{step}.
\begin{figure}[hbtp]
\centering
\includegraphics[height=.7in]{step}
\caption{An ornamental frieze pattern with the glide reflection symmetry.}
\label{step}
\end{figure}
\item The entries of a frieze pattern are given by the 3-diagonal determinants
$$
\det\left|\begin{array}{cccccc}
a_{j}&1&&&\\\
1&a_{j+1}&1&&\\
&\ddots&\ddots&\ddots&\\
&&1&a_{i-1}&1\\
&&&1&a_{i}
\end{array}\right|,
$$
the continuants (called so because of their relation with continued fractions; see \cite{CO} for an intriguing history of this name).
\end{enumerate}
\section{A problem, a theorem, and a counter-example}
I shall be concerned with frieze patterns whose entries are positive real numbers. Given two such frieze patterns of the same width $w$, choose a row and consider the $n$-periodic sequence of the differences of the respective entries of the two friezes. I am interested in the number of sign changes in this sequence over the period.
More precisely, let $1\le k \le [w/2]$ be the number of a row (we don't need to go beyond $[w/2]$ due to the glide symmetry), and let
$v_{i,i+k+1}$ and $u_{i,i+k+1}$ be the entries of $k$th rows of the two frieze patterns. I am interested in the sign changes of $v_{i,i+k+1}-u_{i,i+k+1}$ as $i$ increases by 1 (not excluding the case when either of these differences vanishes).
\medskip
\noindent {\bf Problem 1.}
{\it
For which $k$ must the cyclic sequence $v_{i,i+k+1}-u_{i,i+k+1}$ have at least four sign changes?
}
\medskip
As a partial answer, one has
\medskip
\noindent {\bf Theorem.}
{\it
Four sign changes must occur for $k=1$ and for $k=2$. In addition, for every $k$, four sign changes must occur in the
infinitesimal version of the problem.
}
\medskip
Let me explain the last statement.
Consider a frieze pattern whose first row is constant: $a_i=2x$ for all $i$. Then each next row is constant as well, and their entries, denoted by $U_k(x)$, satisfy $U_{k+1} = 2x U_k (x)- U_{k-1}(x)$ with $U_0(x)=1, U_1(x)=2x$. That is, $U_k(x)$ is the Chebyshev polynomial of the second kind:
$$
U_k(\cos \alpha) = \frac{\sin(k+1)\alpha}{\sin \alpha}.
$$
For this constant frieze pattern to have width $n-3$, set $\alpha=\pi/n$.
For the infinitesimal version of Problem 1, take this constant frieze pattern and its infinitesimal deformation in the class of frieze patterns.
Originally, I hoped that Problem 1 had an affirmative answer for all values of $k$. However, this conjecture was over-optimistic. The following counter-example is provided by Michael Cuntz; in this example, $w=5$ (the smallest possible not to contradict Theorem), all entries are positive rational numbers, and the differences of the entries of the third row are all positive (this row is 4-periodic due to the glide symmetry). I present only the first lines of the two frieze patterns; these are 8-periodic sequences:
$$
\left(2,\ 2,\ 4,\ 2,\ 3,\ \frac{18}{41},\ 41, \frac{30}{41}\right) \ \ {\rm and}\ \
\left( 5,\ \frac{21}{97},\ 194,\ \frac{36}{97},\ 3,\ 5,\ 1,\ 5 \right).
$$
It still may be possible that the bold conjecture holds for Conway-Coxeter frieze patterns that consist of positive integers.
\section{Proofs}
\paragraph{Case $k=1$.}
Let $a_i$ and $b_i$ be the entries of the first rows of the two frieze patterns. Consider the respective discrete Hill's equations
$$
V_{i+1} = a_i V_i - V_{i-1},\ U_{i+1} = b_i U_i - U_{i-1}.
$$
Let $U_i$ and $V_i$ be some solutions.I claim that the sequence $a_i-b_i$ is $\ell_2$-orthogonal to $U_i V_i$:
\begin{equation} \label{orth}
\sum _1^n (a_i - b_i) U_i V_i = 0.
\end{equation}
Indeed,
$$
\sum _1^n (a_i - b_i) U_i V_i = \sum _1^n [U_{i+1} + U_{i-1}] V_i - U_i [V_{i+1} + V_{i-1}] = 0,
$$
due to antiperiodicity.
Note that the space of solutions of a discrete Hill equation is 2-dimensional, and that its solutions are non-oscillating in the sense that they change sign only once over the period (since the entries of the frieze pattern are positive).
Assume that $a_i - b_i$ does not change sign at all. Choose the initial conditions for solutions $U_{i}$ and $V_i$ as follows:
$$
U_1=-1,U_2=1,V_1=-1,V_2=1.
$$
That is, both solution change sign from $i=1$ to $i=2$, and then, due to the non-oscillating property, there are no other sign changes. Hence $U_i V_i >0$ for all $i$, contradicting (\ref{orth}).
Likewise, if $a_i - b_i$ changes sign only twice, from $i_1$ to $i_1+1$, and from $i_2$ to $i_2+1$, choose the initial conditions for solutions $U_i$ and $V_i$ as follows:
$$
U_{i_1}=-1,U_{i_1+1}=1,V_{i_2}=-1,V_{i_2+1}=1.
$$
Then $(a_i - b_i) U_i V_i$ has a constant sign, again contradicting (\ref{orth}).
$\Box$\bigskip
This result, along with its proof, is a discrete version of the following theorem from \cite{OT1} concerning Hill's equations $\varphi''(x) = k(x) \varphi(x)$ whose solutions are $\pi$-antiperiodic (and hence the potential $k(x)$ is $\pi$-periodic) and disconjugate, meaning that every solution changes sign only once on the period $[0,\pi)$. The claim is that, {\it given two such equations, the function $k_1(x) - k_2(x)$ has at least four zeroes on $[0,\pi)$}.
This theorem is equivalent to the beautiful theorem of E. Ghys: {\it the Schwartzian derivative of a diffeomorphism of ${\mathbb {RP}}^1$ has at least four distinct zeroes}, see \cite{OT3} for the relation of the Schwartzian derivative with the Hill equation, and an explanation why zeroes of the Schwartzian derivative are the vertices of a curve in Lorentzian geometry.
\paragraph{Case $k=2$.}
As I mentioned, to a frieze pattern there corresponds an $n$-gon in ${\mathbb {RP}}^1$. The entries of the second row of the frieze pattern are the cross-ratios of the consecutive quadruples of the vertices of this $n$-gon, where cross-ratio is defined as
$$
[a,b,c,d]_1 = \frac{(d-a)(c-b)}{(d-c)(b-a)},
$$
see \cite{Mor}.
On the other hand, one of the results in \cite{OT2}, another discretization of Ghys's theorem, states that, given two cyclically ordered $n$-tuples of points $x_i$ and $y_i$ in ${\mathbb {RP}}^1$, the difference of the cross-ratios $[x_i,x_{i+1},x_{i+2},x_{i+3}]_2 - [y_i,y_{i+1},y_{i+2},y_{i+3}]_2$ changes sign at least four times; here the cross-ratio is defined by
$$
[a,b,c,d]_2 = \frac{(d-b)(c-a)}{(d-c)(b-a)}.
$$
To complete the proof, observe that $[a,b,c,d]_2 - [a,b,c,d]_1 =1$.
$\Box$\bigskip
\paragraph{Infinitesimal version, $k$ arbitrary.}
Consider the polygonal line
$$
V_i = \frac{1}{\sqrt{\sin \frac{\pi}{n}}} \left(\cos \frac{i\pi}{n}, \sin \frac{i\pi}{n} \right),
$$
so that $[V_i,V_{i+1}]=1$ and $V_{i+n}=-V_i$ hold.
Let
$$
W_i = V_i + {\varepsilon} E_i,\ [W_i,W_{i+1}]=1
$$
be an infinitesimal deformation of this polygon $V_i$. I assume in our calculations that ${\varepsilon}^2=0$.
Let
\begin{equation} \label{EtoV}
E_i=p_iV_i+\bar p_i V_{i+1} = q_i V_i + \bar q_i V_{i-1}.
\end{equation}
We shall express the $n$-periodic coefficients $p_i,\bar p_i, \bar q_i$ via the coefficients $q_i$, that solely determine the deformation.
To do so, use the fact that $V_{i+1}=cV_i-V_{i-1}$ with $c= 2 \cos (\pi/{n})$.
This linear relation must be equivalent to the second equality in (\ref{EtoV}), hence
$$
q_i-p_i=c \bar p_i,\ \bar p_i = - \bar q_i.
$$
We also have $[W_i,W_{i+1}]=1$, implying $[V_i,E_{i+1}]+[E_i,V_{i+1}]=0$ and, using (\ref{EtoV}),
$p_i=-q_{i+1}$. Thus
\begin{equation} \label{viaq}
p_i = -q_{i+1},\ \bar p_i = \frac{1}{c} (q_i+q_{i+1}),\ \bar q_i = -\frac{1}{c} (q_i+q_{i+1}).
\end{equation}
Now fix $k$ and consider the deformation of $(k-1)$st row of the frieze pattern:
$$
[W_i,W_{i+k}]=[V_i,V_{i+k}] + {\varepsilon} ([V_i,E_{i+k}]+[E_i,V_{i+k}]).
$$
Using (\ref{EtoV}) and (\ref{viaq}), one finds
\begin{equation*}
\begin{split}
[V_i,E_{i+k}]+[E_i,V_{i+k}] = (q_{i+k}-q_{i+1}) \left([V_i,V_{i+k}] - \frac{1}{c} [V_i,V_{i+k-1}]\right)&\\
- \frac{1}{c} (q_{i+k+1}-q_i) [V_i,V_{i+k-1}]&\\
= \frac{1}{\sin \frac{2\pi}{n}} \left[ (q_{i+k}-q_{i+1}) \sin \frac{\pi (k+1)}{n} - (q_{i+k+1}-q_i) \sin \frac{\pi (k-1)}{n} \right]&.
\end{split}
\end{equation*}
We want to show that the sequence
\begin{equation} \label{test}
c_i := (q_{i+k}-q_{i+1}) \sin \frac{\pi (k+1)}{n} - (q_{i+k+1}-q_i) \sin \frac{\pi (k-1)}{n}
\end{equation}
must change sign at least four times.
First, observe that $c_i$ is $\ell_2$-orthogonal to the constant sequence $(1,\ldots,1)$, that is, $\sum_{i=1}^n c_i =0$; hence $c_i$ must have sign changes.
Next, I claim that $c_i$ is $\ell_2$-orthogonal to the sequence $\sin (2\pi i/n)$.
Indeed
\begin{equation*}
\begin{split}
\sum_{i=1}^{n} c_i \sin \frac{2\pi i}{n} = \sum_{i=1}^{n} &q_i \sin \frac{\pi (k+1)}{n}
\left( \sin \frac{2\pi (i-k)}{n} - \sin \frac{2\pi (i-1)}{n} \right)\\
- &q_i \sin \frac{\pi (k-1)}{n} \left( \sin \frac{2\pi (i-k-1)}{n} + \sin \frac{2\pi i}{n} \right).
\end{split}
\end{equation*}
Hence twice the coefficient of $q_i$ on the right hand side equals
\begin{equation*}
\begin{split}
&\sin \frac{\pi (k+1)}{n} \sin \frac{\pi (1-k)}{n} \cos \frac{\pi (2i-k-1)}{n}\\
&+ \sin \frac{\pi (k-1)}{n} \sin \frac{\pi (1+k)}{n} \cos \frac{\pi (2i-k-1)}{n} =0,
\end{split}
\end{equation*}
as needed.
Similarly, $c_i$ is $\ell_2$-orthogonal to the sequence $\cos (2\pi i/n)$.
Finally, if $c_i$ changes sign only twice, one can find a linear combination
$$
c+ a \sin \frac{2\pi i}{n} + b \cos \frac{2\pi i}{n},
$$
a discrete first harmonic, that changes sign at the same positions as $c_i$. This ``first harmonic" has no other sign changes, so its signs coincide with those of $c_i$. But it is also orthogonal to $c_i$, a contradiction.
$\Box$\bigskip
\section{Back to four vertices, and another problem}
Perhaps the oldest result in the spirit of the four vertex-like theorem is the Legendre-Cauchy Lemma (which is about 100 years older than the theorem of Mukhopadhyaya):
{\it if two convex polygons in the plane have equal respective side length, then the cyclic sequence of the differences of their respective angles has at least four sign changes}.
A version of this lemma in spherical geometry is the main ingredient of the proof of the Cauchy rigidity theorem ({\it convex polytopes with congruent corresponding faces are congruent to each other}); interestingly, its original proof contained an error that remained unnoticed for nearly a century, see, e.g., chapters 22 and 26 of \cite{Pak}.
The values of the angles in the formulation of the Legendre-Cauchy Lemma can be replaces by the lengths of the short, skip-a-vertex, diagonals of the respective polygons: with fixed side lengths, the angles depend monotonically on these diagonals.
In particular, one may assume that the polygons are equilateral, e.g., each side has unit length. In this formulation, the Legendre-Cauchy Lemma becomes an analog of the $k=1$ case of Theorem above, with the determinants $a_i=\det (V_{i-1},V_{i+1})$ replaced by the lengths $|V_{i+1}-V_{i-1}|$. This prompts to ask another question.
\medskip
\noindent {\bf Problem 2.}
{\it
Given two equilateral convex $n$-gons, for which $k$ must the cyclic sequence $|V_{i+k}-V_{i-1}|$ have at least four sign changes?
}
\bigskip
{\bf Acknowledgements}. It is a pleasure to acknowledge stimulating discussions with S. Morier-Genoud, V. Ovsienko, I. Pak, and R. Schwartz.
Many thanks to M. Cuntz for providing his (counter)-examples. This work was supported by NSF grant DMS-1510055.
| {'timestamp': '2018-06-18T02:10:56', 'yymm': '1805', 'arxiv_id': '1805.12065', 'language': 'en', 'url': 'https://arxiv.org/abs/1805.12065'} | ArXiv |
\section{Introduction}
This paper considers the question of determining the number of covers
between genus-$0$ curves with fixed ramification in positive
characteristic. More concretely, we consider covers $f:{\mathbb P}^1\to
{\mathbb P}^1$ branched at $r$ ordered points $Q_1, \ldots, Q_r$ of fixed {\em
ramification type} $(d; C_1, \ldots, C_r)$, where $d$ is the degree of
$f$ and $C_i=e_1(i)\text{-}\cdots\text{-}e_{s_i}(i)$ is a conjugacy
class in $S_d$. This notation indicates that there are $s_i$
ramification points in the fiber $f^{-1}(Q_i)$, with ramification
indices $e_j(i)$. The {\em Hurwitz number} $h(d; C_1, \ldots, C_r)$
is the number of covers of fixed ramification type over ${\mathbb C}$, up to
isomorphism. This number does not depend on the position of the branch
points. If $p$ is a prime not dividing any of the ramification
indices $e_j(i)$, the {\em $p$-Hurwitz number} $h_p(d; C_1, \ldots,
C_r)$ is the number of covers of fixed ramification type whose branch
points are generic over an algebraically closed field $k$ of
characteristic $p$. The genericity hypothesis is necessary because in
positive characteristic the number of covers often depends on
the position of the branch points.
The only general result on $p$-Hurwitz numbers is that they are always
less than or equal to the Hurwitz number, with equality when the degree
of the Galois closure is prime to $p$. This is because every tame
cover in characteristic $p$ can be lifted to characteristic $0$,
and in the prime-to-$p$ case, every cover in characteristic $0$ specializes
to a cover in characteristic $p$ with the same ramification type
(see Corollaire 2.12 of Expos\'e XIII in \cite{sga1}). We say a cover
has {\em good reduction} when
such a specialization exists. However, in the general case, some covers in
characteristic $0$ specialize to inseparable covers in characteristic $p$;
these covers are said to have {\em bad
reduction}. Thus, the difference $h(d;C_1,\dots,C_r)-h_p(d;C_1,\dots,C_r)$
is the number of covers in characteristic $0$ with generic branch points and
bad reduction. In \cite{os7} and \cite{os12}, the value
$h_p(d; e_1, e_2, e_3)$ is computed for genus $0$ covers and any $e_i$ prime
to $p$ using linear series techniques. In this paper, we treat the
considerably more difficult case of genus-$0$ covers of type
$(p; e_1, e_2, e_3, e_4)$. Our main result is the following.
\begin{thm}\label{thm:main} Given $e_1,\dots,e_4$ all less than $p$,
with $\sum_i e_i=2p+2$, we have
\[
h_p(p; e_1, e_2, e_3, e_4)=h(p; e_1, e_2, e_3, e_4)-p.
\]
\end{thm}
An important auxiliary result is the computation of the
$p$-Hurwitz number $h_p(p; e_1\text{-}e_2, e_3, e_4)$.
\begin{thm}\label{thm:3-hurwitz} Given odd integers $e_1 , e_2, e_3, e_4 < p$,
with $e_1 + e_2 \leq p$
and $\sum_i e_i=2p+2$, we have that
\[
h_p(p; e_1\text{-}e_2, e_3, e_4)=
\begin{cases}h(p; e_1\text{-}e_2, e_3, e_4)-(p+1-e_1-e_2): & e_1 \neq e_2, \\
h(p; e_1\text{-}e_2, e_3, e_4)-(p+1-e_1-e_2)/2: & e_1 =e_2.
\end{cases}
\]
\end{thm}
Corollary \ref{cor:2cyclebad} gives a more general result including
the case that some of the $e_i$ are even, but in some cases we also compute
the $p$-Hurwitz number only up to a factor $2$. Note that there is an
explicit formula for $h(p;e_1,e_2,e_3,e_4)$ and
$h(p;e_1\text{-}e_2,e_3,e_4)$; see Theorem
\ref{hurwitzlem} and Lemma \ref{lem:badtype} below.
Our technique involves the use of ``admissible covers,'' which are
certain covers between degenerate curves (see Section
\ref{sec:char0}). Admissible covers provide a compactification of the
space of covers of smooth curves in characteristic $0$, but in
positive characteristic this is not the case, and it is an interesting
question when, under a given degeneration of the base, a cover of
smooth curves does in fact have an admissible cover as a limit. In
this case we say the smooth cover has {\em good degeneration}. In
\cite{bo3} one finds examples of covers with generic branch points
without good degeneration.
In contrast, our technique for proving Theorem \ref{thm:main}
simultaneously shows that many of the examples we consider have good
degeneration.
\begin{thm}\label{thm:good-degen} Given odd integers
$1< e_1 \leq e_2 \leq e_3 \leq e_4<p$ with $\sum_i e_i=2p+2$, every cover
of type $(p;e_1,e_2,e_3,e_4)$ with generic branch points $(0,1,\lambda,\infty)$
has good degeneration under the degeneration sending $\lambda$ to $\infty$.
\end{thm}
As with Theorem \ref{thm:3-hurwitz}, our methods do not give a complete
answer in some cases with even $e_i$, but we do prove a more general
result in Theorem \ref{thm:main2}.
Building on the work of Raynaud \cite{ra3}, Wewers uses the theory of
stable reduction in \cite{we1} to give formulas for the number of
covers with three branch points and having Galois closure of
degree strictly divisible by $p$ which have bad reduction to
characteristic $p$. In \cite{b-w6}, some $p$-Hurwitz numbers are
calculated using the existence portion of Wewers' theorems, but these
are in cases which are rigid (meaning the classical Hurwitz number is
$1$) or very close to rigid, so one does not have to carry out
calculations with Wewers' formulas. In \cite{se6}, Selander uses the
full statement of Wewers' formulas to compute some examples in small
degree. Our result in Theorem \ref{thm:3-hurwitz} is the first
explicit calculation of an infinite family of $p$-Hurwitz numbers
which fully uses Wewers' formulas, and its proof occupies the bulk of
the present paper.
We begin in Sections \ref{sec:char0} and \ref{sec:group} by reviewing the
situation in characteristic $0$ and some group-theoretic background. We
then recall the theory of stable reduction in Section \ref{sec:stable}.
In order to apply Wewers' formulas, in Section \ref{sec:tail} we analyze
the possible structures of the stable reductions which arise, and then
in Section \ref{sec:3pt} we
apply Wewers' formulas to compute the number of smooth covers with a
given stable reduction. Here we are forced to use a trick comparing
the number of covers in the case of interest to the number in a
related case where we know all covers have bad reduction.
In Section \ref{sec:adm} we then apply Corollary \ref{cor:2cyclebad} as
well as the formulas
for $h_p(d;e_1,e_2,e_3)$ of \cite{os7} and the classical Hurwitz
number calculations in \cite{o-l2} to estimate the number of
admissible covers in characteristic $p$. This provides a sufficient
lower bound on $h_p(p;e_1,e_2,e_3,e_4)$. Finally, we use the
techniques of \cite{bo4}, again based on stable reduction, to directly
prove in Section \ref{sec:4pt} that $h_p(p;e_1,e_2,e_3,e_4)$ is bounded above
by $h(p;e_1,e_2,e_3,e_4)-p$. We thus conclude Theorems \ref{thm:main} and
\ref{thm:good-degen}.
We would like to thank Peter M\"uller, Bj\"orn Selander and Robert
Guralnick for helpful discussions.
\section{The characteristic-$0$ situation}\label{sec:char0}
In this paper, we consider covers $f:{\mathbb P}^1\to {\mathbb P}^1$ branched at $r$
ordered points $Q_1, \ldots, Q_r$ of fixed {\em ramification type}
$(d; C_1, \ldots, C_r)$, where $d$ is the degree of $f$ and
$C_i=e_1(i)\text{-}\cdots\text{-}e_{s_i}(i)$ is a conjugacy class in
$S_d$. This means that there are $s_i$ ramification points in the
fiber $f^{-1}(Q_i)$, with ramification indices $e_j(i)$. The {\em
Hurwitz number} $h(d; C_1, \ldots, C_r)$ is the number of covers of
fixed ramification type over ${\mathbb C}$, up to isomorphism. This number
does not depend on the position of the branch points.
Riemann's Existence Theorem implies that the Hurwitz number $h(d; C_1,
\ldots, C_r)$ is the cardinality of the set of {\em Hurwitz
factorizations} defined as
\[
\{(g_1, \cdots, g_r)\in C_1\times \cdots \times C_r\mid \langle
g_i\rangle\subset S_d\, {\rm transitive },\, \prod_i g_i=1\}/\sim,
\]
where $\sim$ denotes uniform conjugacy by $S_d$.
The group $\langle g_i \rangle$ is called the {\em monodromy group}
of the corresponding cover. For a fixed monodromy group $G$, a variant
equivalence relation is given by {\em $G$-Galois covers}, where we work
with Galois covers together with a fixed isomorphism of the Galois group
to $G$.
The group-theoretic interpretation is then that the $g_i$ are in $G$ (with
the action on a fiber recovered by considering $G$ as a subgroup of
$S_{|G|}$), and the equivalence relation $\sim_G$ is uniform conjugacy by $G$.
To contrast with the $G$-Galois case, we sometimes emphasize
that we are working up to $S_d$-conjugacy by referring to the
corresponding covers as {\em mere covers}.
In this paper, we are mainly interested in the {\em pure-cycle} case,
where every $C_i$ is the
conjugacy class in $S_d$ of a single cycle. In this case, we write
$C_i= e_i$, where $e_i$ is the length of the cycle. A cover $f:Y\to
{\mathbb P}^1$ over ${\mathbb C}$ of ramification type $(d; e_1, e_2, \cdots, e_r)$
has genus $g(Y)=0$ if and only if $\sum_{i=1}^r e_i=2d-2+r$.
Giving closed formulae for Hurwitz numbers may get very complicated,
even in characteristic zero. The following result from \cite{o-l2}
illustrates that the genus-$0$ pure-cycle case is more tractable than
the general case, as one may give closed formulae for the Hurwitz numbers,
at least if the number $r$ of branch points is at most $4$.
\begin{thm}\label{hurwitzlem} Under the hypothesis $\sum_{i=1}^r e_i=2d-2+r$,
we have the following.
\begin{itemize}
\item[(a)] $h(d; e_1, e_2, e_3)=1$.
\item[(b)] $h(d; e_1, e_2, e_3, e_4)=\min_i(e_i(d+1-e_i)).$
\item[(c)] Let $f:{\mathbb P}^1_{\mathbb C}\to {\mathbb P}^1_{\mathbb C}$ be a cover of ramification
type $(d; e_1, e_2, \ldots, e_r)$ with $r\geq 3$. The Galois group of
the Galois closure of $f$ is either $S_d$ or $A_d$ unless $(d; e_1,
e_2, \ldots, e_r)=(6; 4,4,5)$ in which case the Galois group is $S_5$
acting transitively on $6$ letters.
\end{itemize}
\end{thm}
These statements are Lemma 2.1, Theorem 4.2, and Theorem 5.3 of
\cite{o-l2}.
We mention that Boccara (\cite{bo5}) proves a partial generalization of
Theorem \ref{hurwitzlem}.(a). He gives a necessary and sufficient condition
for $h(d; C_1, C_2, \ell)$ to be nonzero in the case that $C_1, C_2$
are arbitrary conjugacy classes of $S_d$ and only $C_3=\ell$ is
assumed to be the conjugacy class of a single cycle.
Later in our analysis we will be required to study covers of
type $(d;e_1\text{-}e_2, e_3,e_4)$, so we mention a result which
is not stated explicitly in \cite{o-l2},
but which follows easily from the arguments therein. We will only use
the case that $e_4=d$, but we state the result in general since the
argument is the same.
\begin{lem}\label{lem:badtype}
Given $e_1,e_2,e_3,e_4$ and $d$ with $2d+2=\sum_i e_i$ and
$e_1 + e_2 \leq d$, if $e_1 \neq e_2$ we have
\[
h(d;e_1\text{-}e_2, e_3, e_4)= (d+1-e_1-e_2)\min(e_1,e_2,d+1-e_3,d+1-e_4),
\]
and if $e_1=e_2$ we have
\[
h(d;e_1\text{-}e_2, e_3, e_4)=
\lceil\frac{1}{2}(d+1-e_1-e_2)\min(d+1-e_3,d+1-e_4)\rceil.
\]
Note that this number is always positive.
In particular, when $e_4=d$ we have
\[
h(d;e_1\text{-}e_2, e_3, d)=
\begin{cases} d+1-e_1-e_2& \text{ if }e_1\neq e_2,\\
(d+2-e_1-e_2)/2&\text{ if }e_1=e_2, d \text{ even},\\
(d+1-e_1-e_2)/2&\text{ if }e_1=e_2, d \text{ odd}.
\end{cases}
\]
\end{lem}
\begin{proof} Without loss of generality, we may assume that
$e_1\leq e_2$ and $e_3 \leq e_4$. Thus, we want to prove that
$h(d;e_1\text{-}e_2, e_3, e_4)$ is given by the smaller of
$e_1(d+1-e_1-e_2)$ and $(d+1-e_4)(d+1-e_1-e_2)$
when $e_1\neq e_2$, by $((d+1-e_4)(d+1-e_1-e_2)+1)/2$ when $e_1=e_2$
and all of $d,e_3,e_4$ are even, and by $(d+1-e_4)(d+1-e_1-e_2)/2$
otherwise. Even though we do not assume $e_2 \leq e_3$, this formula
still follows from
the argument of Theorem 4.2.(ii) of \cite{o-l2}.
The first observations to make are that since $e_1+e_2 \leq d$,
we have $e_3+e_4 \geq d+2$, and it follows that although we may not have
$e_2 \leq e_3$, we have $e_1 < e_4$. Moreover, we have $e_1+e_3 \leq d+1$
and $e_2+e_4 \geq d+1$. We are then able to check that the Hurwitz
factorizations $(\sigma_1, \sigma_2, \sigma_3, \sigma_4)$ described in
case (ii) of {\it loc.\ cit.}\ still give valid Hurwitz factorizations
$(g_1,g_2,g_3)$ by setting $g_1=\sigma_1\sigma_2$, just as in Proposition
4.7 of {\it loc.\ cit.} Moreover, just as in Proposition 4.9 of
{\it loc.\ cit.} we find that every Hurwitz factorization must be one
of the enumerated ones.
It remains to consider when two of the described possibilities yield
equivalent Hurwitz factorizations. If $e_1\neq e_2$, we can extract
$\sigma_1$ and $\sigma_2$ as the disjoint cycles (of distinct orders)
in $g_1$, so we easily see that the proof of Proposition 4.8 of
{\it loc. cit.} is still valid. Thus the Hurwitz number is simply the
number of possibilities enumerated in Theorem 4.2 (ii) of \cite{o-l2},
which is the minimum of $e_1(d+1-e_1-e_2)$ and $(d+1-e_4)(d+1-e_1-e_2)$,
as desired.
Now suppose $e_1=e_2$. We then check easily that $e_1+e_4 \geq d+1$, so
that the number of enumerated possibilities is $(d+1-e_4)(d+1-e_1-e_2)$.
Here, we see that we potentially have a given Hurwitz factorization
$(g_1,g_2,g_3)$ being simultaneously equivalent to two of the enumerated
possibilities, since $\sigma_1$ and $\sigma_2$ can be switched. Indeed,
the argument of Proposition 4.8 of {\it loc.\ cit.}\ describing how to
intrinsically recover the parameters $m,k$ of Theorem 4.2 (ii) of
{\it loc.\ cit.}\ lets us compute how $m,k$
change under switching $\sigma_1$ and $\sigma_2$, and we find that
the pair $(m,k)$ is sent to $(e_3+2e_4-d-m, e_3+e_4-d-k)$. We thus find
that each Hurwitz factorization is equivalent to two distinct enumerated
possibilities, with the exception that if $d$ and $e_4$ (and therefore
necessarily $e_3$) are even, the Hurwitz factorization corresponding to
$(m,k)=((e_3+2e_4-d)/2,(e_3+e_4-d)/2)$ is not equivalent to any other.
We therefore conclude the desired statement.
\end{proof}
We now explain how Theorem 4.2 of \cite{o-l2} can be understood in
terms of degenerations. Harris and Mumford \cite{h-m2} developed the
theory of {\em admissible covers}, giving a description of the
behavior of branched covers under degeneration.
Admissible covers in the case we are interested in may be described
geometrically as follows: let $X_0$ be the reducible curve consisting
of two smooth rational components $X_0^1$ and $X_0^2$ joined at a
single node $Q$. We suppose we have points $Q_1,Q_2$ on $X_0^1$, and
$Q_3,Q_4$ on $X_0^2$. An {\em admissible cover} of type
$(d;C_1,C_2,\ast,C_3,C_4)$ is then a connected, finite flat cover
$f_0:Y_0 \to X_0$ which is \'etale away from the preimage of $Q$ and
the $Q_i$, and if we denote by $Y_0^1\to X_0^1$ and $Y_0^2\to X_0^2$ the
(possibly disconnected) covers of $X_0^1$ and $X_0^2$ induced by $f_0$,
we require also that $Y_0^1\to X_0^1$
has ramification type $(d;C_1,C_2,C)$ for $Q_1,Q_2,Q$ and $Y_0^2\to
X_0^2$ has ramification type $(d;C,C_3,C_4)$ for $Q,Q_3,Q_4$, for some
conjugacy class $C$ in $S_d$, and furthermore that for $P \in f_0^{-1}(Q)$,
the ramification index of $f_0$ at $P$ is the same on $Y_0^1$ and $Y_0^2$.
In characteristic $p$, we further have to require that ramification above
the node is tame. We refer to $Y_0^1 \to X_0^1$ and $Y_0^2 \to X_0^2$ as
the {\em component covers} determining $f_0$. When we wish to specify the
class $C$, we say the admissible cover is of type
$(d,C_1,C_2,\ast_C,C_3,C_4)$.
The two
basic theorems on admissible covers concern degeneration and smoothing.
First, in characteristic $0$, or when the monodromy group has order prime to
$p$, if a family of smooth covers of type $(d;C_1,C_2,C_3,C_4)$ with branch
points $(Q_1,Q_2,Q_3,Q_4)$ is degenerated by sending $Q_3$ to $Q_4$, the
limit is an admissible cover of type $(d;C_1,C_2,\ast,C_3,C_4)$. On the
other hand, given an admissible cover of type $(d;C_1,C_2,\ast,C_3,C_4)$,
irrespective of characteristic there is a deformation to a cover of smooth
curves,
which then has type $(d;C_1,C_2,C_3,C_4)$. Such a deformation is not unique
in general; we call the number of smooth covers arising as smoothings of
a given admissible cover (for a fixed smoothing of the base) the
{\em multiplicity} of the admissible cover.
Suppose we have a family of covers $f:X \to Y$, with smooth generic fiber
$f_1:X_1 \to Y_1$, and admissible special fiber $f_0:X_0 \to Y_0$.
If we choose local monodromy generators for
$\pi_1^{\operatorname{tame}}(Y_1 \smallsetminus \{Q_1,Q_2,Q_3,Q_4\})$ which are
compatible with the degeneration to $Y_0$, we then find that if we have
a branched cover of $Y_1$ corresponding to a Hurwitz factorization
$(g_1,g_2,g_3,g_4)$, the induced admissible cover
of $Y_0$ will have monodromy given by $(g_1,g_2,\rho)$ over
$Y_0^1$ and $(\rho^{-1},g_3,g_4)$ over $Y_0^2$, where
$\rho=g_3 g_4$. The multiplicity of the admissible cover arises
because it may be possible to apply different simultaneous conjugations
to $(g_1,g_2,\rho)$ and to $(\rho^{-1},g_3,g_4)$
while maintaining the relationship between $\rho$ and $\rho^{-1}$.
It is well-known that when $\rho$ is a pure-cycle of order $m$, the
admissible cover has multiplicity $m$, although we recover this fact
independently in our situation as part of the Hurwitz number calculation
of \cite{o-l2}.
To calculate more generally the multiplicity of an admissible cover of
the above type, we define the action of the braid operator $Q_3$ on the
set of Hurwitz factorizations as
\[
Q_3\cdot (g_1, g_2, g_3, g_4)=(g_1g_2g_1g_2^{-1}g_1^{-1}, g_1
g_2g_1^{-1}, g_3, g_4).
\]
One easily checks that $Q_3\cdot\bar{g}$ is again a Hurwitz
factorization of the same ramification type as $\bar{g}$. The multiplicity
of a given admissible cover is the length of the orbit of $Q_3$ acting
on the corresponding Hurwitz factorization.
In this context, we can give the following sharper statement of
Theorem \ref{hurwitzlem} (b), phrased in somewhat different language
in \cite{o-l2}.
\begin{thm}\label{degenerationlem}
Given a genus-$0$ ramification type $(d; e_1, e_2, e_3, e_4)$, with
$e_1 \leq e_2 \leq e_3 \leq e_4$
the only possibilities for an admissible cover of type
$(d;e_1,e_2,\ast,e_3,e_4)$ are type
$(d;e_1,e_2,\ast_{m},e_3,e_4)$ or type
$(d;e_1,e_2,\ast_{e_1\text{-}e_2},e_3,e_4)$.
\begin{itemize}
\item[(a)] Fix $m \geq 1$. There is at most one admissible cover
of type $(d;e_1,e_2,\ast_{m},e_3,e_4)$, and if such a cover exists,
it has multiplicity $m$.
\begin{itemize}
\item[(i)] Suppose that $d+1\leq e_2+e_3$.
There exists an admissible cover of type $(d;e_1,e_2,\ast_{m},e_3,e_4)$
if and only if
\[
e_2-e_1+1\leq m\leq 2d+1-e_3-e_4, \qquad m\equiv e_2-e_1+1
\pmod{2}.
\]
\item[(ii)] Suppose that $d+1\geq e_2+e_3$.
There exists an admissible cover of type $(d;e_1,e_2,\ast_{m},e_3,e_4)$
if and only if
\[
e_4-e_3+1\leq m\leq
2d+1-e_3-e_4,\qquad m\equiv e_2-e_1+1 \pmod{2}.
\]
\end{itemize}
\item[(b)] Admissible covers of type
$(d;e_1,e_2,\ast_{e_1\text{-}e_2},e_3,e_4)$
have multiplicity $1$. The component cover of type
$(d;e_1,e_2,e_1\text{-}e_2)$ is uniquely determined, so the admissible
cover is determined by its second component cover and the gluing over
the node. Moreover, the gluing over the node is unique when $e_1 \neq e_2$.
When $e_1=e_2$, there are always two possibilities for gluing except for
a single admissible cover in the case that $e_3,e_4,$ and $d$ are all even.
The number of admissible covers of this type is
\[
\begin{cases}
e_1(d+1-e_1-e_2)& \text{ if }d+1\leq e_2+e_3,\\
(e_3+e_4-d-1)(d+1-e_4)&\text{ if } d+1\geq e_2+e_3.
\end{cases}
\]
\end{itemize}
\end{thm}
\begin{proof} We briefly explain how this follows from Theorem 4.2 of
\cite{o-l2}. As stated above, the possibilities for admissible covers
are determined by pairs $(g_1,g_2,\rho)$, $(\rho^{-1},g_3,g_4)$ where
$(g_1,g_2,g_3,g_4)$ is a Hurwitz factorization of type $(d;e_1,e_2,e_3,e_4)$.
{\it Loc.\ cit.}\ immediately implies that $\rho$ is always either
a single cycle of length $m \geq 1$ or the product of two disjoint
cycles.
For (a), we find from part (i) of {\it loc.\ cit.}\ that the
ranges for $m$ (which is $e_3+e_4-2k$ is the notation of {\it loc.\ cit.})
are as asserted, and that for a given $m$,
the number of possibilities with $\rho$ an $m$-cycle is precisely $m$,
when counted with multiplicity. On the other hand, in this case both
component covers are three-point pure-cycle covers, and thus uniquely
determined (see Theorem \ref{hurwitzlem} (a)). Thus the admissible cover
is unique in this case, with multiplicity $m$.
For (b), we see by inspection of the description of
part (ii) of {\it loc.\ cit.}\ that $g_1$ is disjoint from $g_2$.
It immediately follows that the braid action is trivial, so the
multiplicity is always $1$, and the asserted count of covers follows
immediately from the proof of Proposition 4.10 of {\it loc.\ cit.}
Moreover, the component cover of type $(d;e_1,e_2,e_1\text{-}e_2)$
is a disjoint union of covers of type $(e_1;e_1,e_1)$ and $(e_2;e_2,e_2)$
(as well as $d-e_1-e_2$ copies of the trivial cover), so it is uniquely
determined, as asserted. Furthermore, we see that the second component cover
is always a single connected cover of degree $d$, and $g_1,g_2$ are
recovered as the disjoint cycles of $\rho^{-1}$, so the gluing is unique
when $e_1 \neq e_2$. When $e_1=e_2$, it is possible to swap $g_1$ and
$g_2$, so we see that there are two possibilities for gluing. The
argument of Lemma \ref{lem:badtype} shows that we do in fact obtain
two distinct admissible covers in this way, except for a single
cover occurring when $e_3,e_4$ and $d$ are all even.
\end{proof}
\section{Group theory}\label{sec:group}
In several contexts, we will have to calculate monodromy groups other
than those treated by Theorem \ref{hurwitzlem} (c). We will also have
to pass between counting mere covers and counting $G$-Galois covers.
In this section, we give basic group-theoretic results to address
these topics.
Since we restrict
our attention to covers of prime degree, the following proposition
will be helpful.
\begin{prop}\label{prop:group}
Let $p$ be a prime number and $G$ a transitive group on $p$
letters. Suppose that $G$ contains a pure cycle of length
$1<e < p-2$. Then $G$ is either $A_p$ or $S_p$.
Moreover, if $e=p-2$, and $G$ is neither $A_p$ nor $S_p$, then $p=2^r+1$
for some $r$, and $G$ contains a unique
minimal normal subgroup isomorphic to $\operatorname{PSL}_2(2^r)$, and is contained in
$\operatorname{P\Gamma L}_2(2^r)\simeq \operatorname{PSL}_2(2^r)\rtimes {\mathbb Z}/r{\mathbb Z}$.
If $e=p-1$, and $G$ is not $S_p$, then $G={\mathbb F}_p \rtimes {\mathbb F}_p^*$.
\end{prop}
Note that this does not contradict the exceptional case $d=6$ and
$G=S_5$ in Theorem \ref{hurwitzlem} (c), since we assume that the
degree $d$ is prime.
\begin{proof}
Since $p$ is prime, $G$ is
necessarily primitive, and a theorem usually attributed to Marggraff
(\cite{l-t1}) then states that $G$ is at least $(p-e+1)$-transitive.
When $e\leq p-2$, we have that $p-e+1\geq 3$.
The $2$-transitive permutation groups have been classified by Cameron
(Section 5 of \cite{ca2}). Specifically, $G$ has a unique minimal
normal subgroup which is either elementary abelian or one of several
possible simple groups. Since $G$ is at least $3$-transitive, one easily
checks that the elementary abelian case is not possible: indeed, one checks
directly that if a subgroup of a $3$-transitive group inside $S_p$ contains
an element of prime order $\ell$, then it is not possible for all its
conjugates to commute with one another.
Similarly, most possibilities
in the simple case cannot be $3$-transitive. If $G$ is not
$S_p$ or $A_p$, then $G$ must have a unique
minimal normal
subgroup $N$ which is isomorphic to a Mathieu group $M_{11}, M_{23}$,
or to $N=\operatorname{PSL}_2(2^r)$. We then have that $G$ is a subgroup of $\operatorname{Aut}(N)$.
For $N=M_{11},M_{23}$, we have $N=G=\operatorname{Aut}(N)$, and it is easy to check that
the Mathieu groups $M_{11}$ and $M_{23}$ do not contain any single
cycles of order less than $p$, for example with the computer algebra
package GAP. Therefore
these cases do not occur. The group $\operatorname{PSL}_2(2^r)$ can only occur if
$p=2^r+1$. In this case, we have that $G$ is a subgroup of
$\operatorname{Aut}(\operatorname{PSL}_2(2^r))= \operatorname{P\Gamma L}_2(2^r)$ and $G$ is at most $3$-transitive,
so we have $e=p-2$, as desired.
Finally, if $e=p-1$, M\"uller has classified transitive permutation
groups containing $(p-1)$-cycles in Theorem 6.2 of \cite{mu3}, and
we see that the only possibility in prime degree other than $S_p$
is ${\mathbb F}_p \rtimes {\mathbb F}_p^*$, as asserted.
\end{proof}
We illustrate the utility of the proposition with:
\begin{cor}\label{3pt-monodromy} Fix $e_1,e_2,e_3,e_4$ with
$2 \leq e_i \leq p$ for each $i$, and
$e_1+e_2 \leq p$. For $p>5$, any genus-$0$ cover of type
$(p;e_1\text{-}e_2,e_3,e_4)$ has monodromy group $S_p$ or $A_p$, with
the latter case occurring precisely when $e_3$ and $e_4$ are odd, and
$e_1+e_2$ is even. For $p=5$, the only exceptional case is type
$(5;2\text{-}2,4,4)$, where the monodromy group is ${\mathbb F}_5 \rtimes {\mathbb F}_5^*$.
\end{cor}
\begin{proof}
Without loss of generality, we assume $e_1\leq e_2$ and $e_3 \leq e_4$.
Applying Proposition \ref{prop:group}, it is clear that
the only possible exception occurs for types with $e_3, e_4 \geq p-2$.
We thus have to treat types $(p;3\text{-}3,p-2,p-2)$,
$(p;2\text{-}4,p-2,p-2)$, $(p;2\text{-}2,p-2,p)$, $(p;2\text{-}3,p-2,p-1)$,
and $(p;2\text{-}2,p-1,p-1)$. The fourth case cannot be exceptional since
$G$ contains both a $(p-2)$-cycle and a $(p-1)$-cycle, and the last case
also is ruled out for $p>5$ because ${\mathbb F}_p \rtimes {\mathbb F}_p^*$ does not
contain a $2$-$2$-cycle.
For the first three cases, we must have that $p=2^r+1$ for
some $r$ and $G$ is a subgroup of $\Gamma:=\operatorname{P\Gamma L}_2(2^r)$. Since $p=2^r+1$
is a Fermat prime number, we have that $r$ is a power of $2$. Moreover,
since $\operatorname{PSL}_2(4)=A_5$ as permutation groups in $S_5$, we may assume
$r \geq 4$. Since $r$ is even,
any element of order $3$ in $\Gamma \cong \operatorname{PSL}_2(2^r) \rtimes {\mathbb Z}/r{\mathbb Z}$
must lie inside $\operatorname{PSL}_2(2^r)$, and a non-trivial element of this group
can fix at most $2$ letters. Thus, in order to contain a
$3\text{-}3$-cycle, we would have to have $6 \leq p=2^r+1 \leq 8$, which
contradicts the hypothesis $r \geq 4$. This rules out the first case. In
the second case, if we square the $2$-$4$-cycle we obtain a $2$-$2$-cycle.
To complete the argument for both the second and third cases it is thus
enough to check directly that if $r>4$, an element of order $2$ cannot
fix precisely $p-4$ letters, ruling out a $2$-$2$-cycle in this case.
It remains only to check directly that $\operatorname{P\Gamma L}_2(16)$ does not contain
a $2$-$2$-cycle, which one can do directly with GAP.
\end{proof}
Because the theory of stable reduction is developed in the $G$-Galois
context, it is convenient to be able to pass back and forth between
the context of mere covers and of $G$-Galois covers.
The following easy result relates the number of mere covers to the number
of $G$-Galois covers in the case we are interested in.
\begin{lem}\label{Gallem}
Let $f:{\mathbb P}^1 \to {\mathbb P}^1$ be a (mere) cover of degree $d$ with monodromy
group $G=A_d$ (respectively, $S_d$). Then the number of $G$-Galois
structures on the Galois closure of $f$ is exactly $2$ (respectively, $1$).
\end{lem}
\begin{proof}
The case that $G=S_d$ is clear, since conjugacy by $S_d$ is then the
same as conjugacy by $G$. Suppose $G=A_d$, and
let $X=\{(g_1, \ldots, g_r)\mid
\prod_i g_i=1, \langle g_i\rangle=d\}$.
Since the centralizer $C_{S_d}(A_d)$ of $A_d$ in $S_d$ is trivial, it
follows that $S_d$ acts freely on $X$, so the number of elements in
$X_f \subseteq X$ corresponding to $f$ as a mere cover is $|S_d|$. Since
the center of $G=A_d$ is trivial, $G$ also acts freely on $X$, and $X_f$
breaks into two equivalence classes of $G$-Galois covers, each of
size $|A_d|$.
\end{proof}
\section{Stable reduction}\label{sec:stable}
In this section, we recall some generalities on stable reduction of
Galois covers of curves, and prove a few simple lemmas as a prelude
to our main calculations. The main references for this section are
\cite{we1} and \cite{bo4}. Since these sources only consider the case
of $G$-Galois covers, we restrict to this situation here as well. Lemma
\ref{Gallem} implies that we may translate results on good or bad
reduction of $G$-Galois covers to results on the stable reduction of the
mere covers, so this is no restriction.
Let $R$ be a discrete valuation ring with fraction field $K$ of
characteristic zero and residue field an algebraically closed field
$k$ of characteristic $p>0$. Let $f:V={\mathbb P}^1_K \to X={\mathbb P}^1_K$ be a
degree-$p$ cover branched at $r$ points $Q_1=0,Q_2=1,\ldots,
Q_r=\infty$ over $K$ with ramification type $(p; C_1, \ldots, C_r).$
For the moment, we do not assume that the $C_i$ are the conjugacy
classes of a single cycle. We denote the Galois closure of $f$ by
$g:Y\to {\mathbb P}^1$ and let $G$ be its Galois group. Note that $G$ is a
transitive subgroup of $S_p$, and thus has order divisible by $p$.
Write $H:={\mathop{\rm Gal}}(Y, V)$, a subgroup of index $p$.
We suppose that $Q_i\not\equiv Q_j\pmod{p}$, for $i\neq j$, in other
words, that $(X; \{Q_i\})$ has good reduction as a marked curve. We
assume moreover that $g$ has bad reduction to characteristic $p$, and
denote by $\bar{g}:\bar{Y}\to \bar{X}$ its {\em stable reduction}. The
stable reduction $\bar{g}$ is defined as follows. After replacing $K$
by a finite extension, there exists a unique stable model ${\mathcal Y}$ of
$Y$ as defined in \cite{we1}. We define ${\mathcal X}={\mathcal Y}/G$. The stable reduction
$\bar{g}:\bar{Y}:={\mathcal Y}\otimes_R k\to \bar{X}:={\mathcal X}\otimes_R k$ is a
finite map between semistable curves in characteristic $p$; we call
such maps {\em stable $G$-maps}. We refer to \cite{we1},
Definition 2.14, for a precise definition.
Roughly speaking, the theory of stable reduction proceeds in two steps:
first, one understands the possibilities for stable $G$-maps, and
then one counts the number of characteristic-$0$ covers reducing
to each stable $G$-map.
We begin by describing what the stable reduction must look like.
Since $(X; Q_i)$ has good reduction to characteristic $p$, there
exists a model ${\mathcal X}_0\to \operatorname{Spec}(R)$ such that the $Q_i$ extend to
disjoint sections. There is a unique irreducible component $\bar{X}_0$
of $\bar{X}$, called the {\em original component}, on which the
natural contraction map $\bar{X}\to {{\mathcal X}}_0\otimes_Rk$ is an
isomorphism. The restriction of $\bar{g}$ to $\bar{X}_0$ is
inseparable.
Let ${\mathbb B} \subseteq \{1,2,\ldots,r\}$ consist of those
indices $i$ such that $C_i$ is not the conjugacy class of a
$p$-cycle. For $i \in {\mathbb B}$, we have that $Q_i$ specializes to an
irreducible component $\bar{X}_i\neq \bar{X}_0$ of $\bar{X}$. The
restriction of $\bar{g}$ to $\bar{X}_i$ is separable, and
$\bar{X}_i$ intersects the rest of $\bar{X}$ in a single point
$\xi_i$. Let $\bar{Y}_i$ be an irreducible component of $\bar{Y}$
above $\bar{X}_i$, and write $\bar{g}_i:\bar{Y}_i\to \bar{X}_i$ for
the restriction of $\bar{g}$ to $\bar{Y}_i$. We denote by $G_i$ the
decomposition group of $\bar{Y}_i$. We call the components
$\bar{X}_i$ (resp.\ the covers $\bar{g}_i$) for $i\in{\mathbb B}$ the {\em
primitive tails} (resp.\ the {\em primitive tail covers}) associated
with the stable reduction. The following definition gives a
characterization of those covers that can arise as primitive tail
covers (compare to \cite{we1}, Section 2.2).
\begin{defn}\label{def:tail} Let $k$ be an algebraically closed field of
characteristic $p>0$. Let $C$ be a conjugacy class of $S_p$ which
is not the class of a $p$-cycle. A {\em primitive tail cover} of
ramification type $C$ is a $G$-Galois cover $\varphi_C:T_C\to {\mathbb P}^1_k$
defined over $k$ which is branched at exactly two points $0, \infty$,
satisfying the following conditions.
\begin{itemize}
\item[(a)] The Galois group $G_C$ of $\varphi_C$ is a subgroup of
$S_p$ and contains a subgroup $H$ of index $p$ such that
$\bar{T}_C:=T_C/H$ has genus $0$.
\item[(b)] The induced map $\bar{\varphi}_C:\bar{T}_C\to {\mathbb P}^1$ is
tamely branched at $x=0$, with conjugacy class $C$, and wildly
branched at $x=\infty$.
\end{itemize}
If $\varphi$ is a tail cover, we let $h=h(\varphi)$ be
the conductor and $pm=pm(\varphi)$ the ramification index of
a wild ramification point.
We say that two primitive tail covers $\varphi_i:T_i\to {\mathbb P}^1_k$ are
{\em isomorphic} if there exists a $G$-equivariant isomorphism $\iota:T_1\to
T_2$. Note that we do not require an isomorphism to send $\bar{T}_1$ to
$\bar{T}_2$.
\end{defn}
Note that an isomorphism $\iota$ of primitive tail covers may be
completed into a commuting square
\[
\begin{CD}
{T}_1 @>{\iota}>> {T}_2\\
@V{{\varphi}_1}VV
@VV{{\varphi}_2}V\\
{\mathbb P}^1 @>>>{\mathbb P}^1.
\end{CD}
\]
Note also that the number of primitive tail covers of fixed
ramification type is finite.
Since $p$ strictly divides the order of the Galois group $G_C$, we
conclude that $m$ is prime to $p$. The invariants $h, m$ describe the
wild ramification of the tail cover $\varphi_C$. The integers $h$ and
$m$ only depend on the conjugacy class $C$. In Section
\ref{sec:tail}, we will show this if $C$ is the class of a single
cycle or the product of $2$ disjoint cycles, but this holds more
generally. In the more general set-up of \cite{we1}, Definition 2.9
it is required that $\sigma:=h/m<1$ as part of the definition of
primitive tail cover. We will see that in our situation this follows from
(a). Moreover, we will show that $\gcd(h, m)=1$ (Lemma
\ref{lem:tail1}). Summarizing, we find that $(h,m)$ satisfy:
\begin{equation}\label{eq:hm}
m\mid (p-1), \qquad 1\leq h < m, \qquad \gcd(h, m)=1.
\end{equation}
In the more general set-up of
\cite{we1} there also exists so-called new tails, which satisfy $\sigma>1$. The
following lemma implies that these do not occur in our situation.
\begin{lem}\label{lem:stablered}
The curve $\bar{X}$ consists of at most $r+1$ irreducible components:
the original component $\bar{X}_0$ and primitive tails $\bar{X}_i$ for
all $i \in {\mathbb B}$.
\end{lem}
\begin{proof}
In the case that $r=3$ this is proved in \cite{we1}, Section 4.4,
using that the cover is the Galois closure of a genus-$0$ cover of
degree $p$. The general case is a straightforward generalization.
\end{proof}
It remains to discuss the restriction of $\bar{g}$ to the original
component $\bar{X}_0$. As mentioned above, this restriction is
inseparable, and it is described by a so-called deformation datum
(\cite{we1}, Section 1.3).
In order to describe deformation data, we set some notation. Let
$\bar{Q}_i$ be the limit on $\bar{X}_0$ of the $Q_i$ for
$i \not \in {\mathbb B}$, and set $\bar{Q}_i=\xi_i$ for $i \in {\mathbb B}$.
\begin{defn}\label{def:dd}
Let $k$ be an algebraically closed field of characteristic $p$. A
{\em deformation datum} is a pair $(Z, \omega)$, where $Z$ is a smooth
projective curve over $k$ together with a finite Galois cover $g:Z\to
X={\mathbb P}^1_k$, and $\omega$ is a meromorphic differential form on $Z$
such that the following conditions hold.
\begin{itemize}
\item[(a)]
Let $H$ be the Galois group of $Z\to X$.
Then
\[\beta^\ast \omega=\chi(\beta)\cdot
\omega, \qquad \mbox{for all } \beta\in H.
\]
Here $\chi:H\to
{\mathbb F}_p^\times$ in an injective character.
\item[(b)] The differential form $\omega$ is either logarithmic,
i.e.\ of the form $\omega={\rm d} f/f$, or exact, i.e.\ of the form
${\rm d} f$, for some meromorphic function $f$ on $Z$.
\end{itemize}
\end{defn}
Note that the cover $Z \to X$ is necessarily cyclic.
To a $G$-Galois cover $g:Y\to {\mathbb P}^1$ with bad reduction, we may
associate a deformation datum, as follows. Choose an irreducible
component $\bar{Y}_0$ of $\bar{Y}$ above the original component
$\bar{X}_0$. Since the restriction $\bar{g}_0:\bar{Y}_0\to\bar{X}_0$
is inseparable and $G\subset S_p$, it follows that
the inertia group $I$ of $\bar{Y}_0$ is cyclic of order $p$, i.e.\ a
Sylow $p$-subgroup of $G$. Since the inertia group is normal in the
decomposition group, the decomposition group $G_0$ of $\bar{Y}_0$ is a
subgroup of $N_{S_p}(I)\simeq {\mathbb Z}/p{\mathbb Z}\rtimes_\chi{\mathbb Z}/p{\mathbb Z}^\ast$,
where $\chi:{\mathbb Z}/p{\mathbb Z}^\ast\to {\mathbb Z}/p{\mathbb Z}^\ast$ is an injective character.
It follows that the map $\bar{g}_0$ factors as $\bar{g}_0:\bar{Y}_0\to
\bar{Z}_0\to \bar{X}_0$, where $\bar{Y}_0\to \bar{Z}_0$ is inseparable
of degree $p$ and $\bar{Z}_0\to\bar{X}_0$ is separable. We conclude
that the Galois group $H_0:=\operatorname{Gal}(\bar{Z}_0, \bar{X}_0)$ is a subgroup of
${\mathbb Z}/p{\mathbb Z}^\ast\simeq {\mathbb Z}/(p-1){\mathbb Z}$. In particular, it follows that
\begin{equation}\label{eq:G0}
G_0\simeq I\rtimes_\chi H_0.
\end{equation}
The inseparable map $\bar{Y}_0\to
\bar{Z}_0$ is characterized by a differential form $\omega$ on
$\bar{Z}_0$ satisfying the properties of Definition \ref{def:dd}, see
\cite{we1}, Section 1.3.2.
The differential form $\omega$ is logarithmic if $\bar{Y}_0\to
\bar{Z}_0$ is a ${\boldsymbol \mu}_p$-torsor and exact if this map is
an ${\boldsymbol \alpha}_p$-torsor. A differential form is logarithmic
if and only if it is fixed by the Cartier operator ${\mathcal C}$ and exact if
and only if it is killed by ${\mathcal C}$. (See for example \cite{g-s2},
exercise 9.6, for the definition of the Cartier operator and an
outline of these properties.) Wewers (\cite{we3}, Lemma 2.8)
shows that in the case of covers branched at $r=3$ points the
differential form is always logarithmic.
The deformation datum $(Z, \omega)$ associated to $g$ satisfies the
following compatibilities with the tail covers. We refer to
\cite{we1}, Proposition 1.8 and (2) for proofs of these
statements. For $i\in {\mathbb B}$, we let $h_i$ (resp.\ $pm_i$) be the
conductor (resp. ramification index) of a wild ramification point of
the corresponding tail cover of type $C_i$, as defined above. We put
$\sigma_i=h_i/m_i$.
We also use the convention $\sigma_i=0$ for $i \not \in {\mathbb B}$.
\begin{itemize}
\item[(a)]
If $C_i$ is the
conjugacy class of a $p$-cycle then $\bar{Q}_i$ is unbranched in
$\bar{Z}_0\to \bar{X}_0$ and $\omega$ has a simple pole at all points
of $\bar{Z}_0$ above $\bar{Q}_i$.
\item[(b)] Otherwise, $\bar{Z}_0\to \bar{X}_0$ is branched of order
$m_i$ at $\bar{Q}_i$, and $\omega$ has a zero of order $h_i-1$ at
the points of $\bar{Z}_0$ above $\bar{Q}_i$.
\item[(c)] The map $\bar{Z}_0\to \bar{X}_0$ is unbranched outside
$\{\bar{Q}_i\}$. All poles and zeros of $\omega$ are above the
$\bar{Q}_i$.
\item[(d)] The invariants $\sigma_i$ satisfy $\sum_{i\in {\mathbb B}} \sigma_i=r-2$.
\end{itemize}
The set $(\sigma_i)$ is called the {\em signature} of the deformation
datum $(Z, \omega)$.
\begin{prop}\label{prop:dd}
Suppose that $r=3, 4$. We fix rational numbers $(\sigma_1, \ldots,
\sigma_r)$ with $\sigma_i\in \frac{1}{p-1}{\mathbb Z}$ and $0\leq\sigma_i< 1$,
and $\sum_{i=1}^r \sigma_i=r-2$. We fix points
$\bar{Q}_1=0, \bar{Q}_2=1, \ldots, \bar{Q}_r=\infty$ on
$\bar{X}_0\simeq {\mathbb P}^1_k$. Then there exists a deformation datum of
signature $(\sigma_i)$, unique up to scaling. If further the $\bar{Q}_i$
are general, the deformation datum is logarithmic and unique up to
isomorphism.
\end{prop}
\begin{proof} In the case that $r=3$ this is proved in \cite{we1}.
(The proof in this case is similar to the proof in the case that
$r=4$ which we give below.) Suppose that $r=4$. Let ${\mathbb B}=\{1\leq
i\leq r\mid \sigma_i\neq 0\}$. We write $\bar{Q}_3=\lambda\in
{\mathbb P}^1_k\setminus\{0,1,\infty\}$ and $\sigma_i=a_i/(p-1)$. (If
$\omega$ is the deformation datum associated with $\bar{g}$, then
$a_i=h_i (p-1)/m_i.$)
It is shown in \cite{bo4}, Chapter 3, that a deformation datum of
signature $(\sigma_i)$ consists of a differential form $\omega$ on the cover
$\bar{Z}_0$ of $\bar{X}_0$ defined as a connected component of the
(normalization of the) projective curve with Kummer equation
\begin{equation}\label{eq:Kummer}
z^{p-1}=x^{a_1}(x-1)^{a_2}(x-\lambda)^{a_3}.
\end{equation}
The degree of $\bar{Z}_0\to \bar{X}_0$ is
\[
m:=\frac{p-1}{\gcd(p-1, a_1, a_2, a_3, a_4)}.
\]
The differential form $\omega$ may be written as
\begin{equation}\label{eq:omega}
\omega=\epsilon\frac{z\,{\rm d}
x}{x(x-1)(x-\lambda)}=
\epsilon \frac{x^{p-a_1}(x-1)^{p-1-a_2}(x-\lambda)^{p-1-a_3}z^p}
{x^p(x-1)^p(x-\lambda)^p}\frac{{\rm d}x}{x},
\end{equation}
where $\epsilon\in k^\times$ is a unit.
To show the existence of the deformation datum, it suffices to show
that one may choose $\epsilon$ such that $\omega$ is logarithmic or
exact, or, equivalently, such that $\omega$ is fixed or killed by the
Cartier operator ${\mathcal C}$. It follows from standard properties of the
Cartier operator, (\ref{eq:omega}), and the assumption that
$a_1+a_2+a_3+a_4=2(p-1)$ that ${\mathcal C}\omega
=c^{1/p}\epsilon^{(1-p)/p}\omega$, where
\begin{equation}\label{eq:Hasseinv}
c=\sum_{j=\max(0, p-1-a_2-a_4)}^{\min(a_4, p-1-a_3)} {p-1-a_2\choose
a_4-j}{p-1-a_3\choose j}\lambda^j.
\end{equation}
Note that $c$ is the coefficient of $x^p$ in
$x^{p-a_1}(x-1)^{p-1-a_2}(x-\lambda)^{p-1-a_3}$. One easily checks
that $c$ is nonzero as polynomial in $\lambda$. It follows that
$\omega$ defines an exact differential form if and only if $\lambda$
is a zero of the polynomial $c$. This does not happen if
$\{0,1,\lambda, \infty\}$ is general.
We assume that $c(\lambda)\neq 0$. Since $k$ is algebraically closed,
we may choose $\epsilon\in k^\times$ such that
$\epsilon^{p-1}=c$. Then ${\mathcal C}\omega=\omega$, and $\omega$ defines a
logarithmic deformation datum. It is easy to see that $\omega$ is
unique, up to multiplication by an element of ${\mathbb F}_p^\times$.
\end{proof}
\section{The tail covers}\label{sec:tail}
In Section \ref{sec:stable}, we have seen that associated with a Galois
cover with bad reduction is a set of (primitive) tail covers. In this
section, we analyze the possible tail covers for conjugacy classes
$e\neq p$ and $e_1$-$e_2$ of $S_p$. Recall from Section \ref{sec:char0} that
these are conjugacy classes which occur in the $3$-point covers obtained
as degeneration of the pure-cycle $4$-point covers.
The following lemma shows the existence of primitive tail covers for the
conjugacy classes occurring in the degeneration of single-cycle
$4$-point covers (Theorem \ref{degenerationlem}).
\begin{lem}\label{lem:tail1}
\begin{itemize}
\item[(a)] Let $2\leq e< p-1$ be an integer. There exists a primitive
tail cover $\varphi_e:T_e\to {\mathbb P}^1_k$ of ramification type $e$. Its
Galois group is $A_p$ if $e$ is odd and $S_p$ if $e$ is even.
The wild branch point
of $\varphi_e$ has inertia group of order $p(p-1)/\gcd(p-1,e-1)
=:pm_{e}$. The conductor is $h_e:=(p-e)/\gcd(p-1, e-1).$
\item[(b)] In the case that $e=p-1$, there exists a primitive tail cover
$\varphi_e$ of ramification type $e$,
with Galois group ${\mathbb F}_p\rtimes {\mathbb F}_p^\ast$. The cover is totally
branched above the wild branch point and has conductor $h_{p-1}=1$.
\item[(c)]
Let $2\leq e_1\leq e_2\leq p-1$ be integers with
$e_1+e_2\leq p$. There is a primitive tail cover
$\varphi_{e_1,e_2}:T_{e_1,e_2}\to {\mathbb P}^1_k$ of ramification type
$e_1$-$e_2$. The wild branch point of $\varphi_{e_1,e_2}$ has inertia
group of order $p(p-1)/\gcd(p-1,e_1+e_2-2)=:pm_{e_1, e_2} $. The conductor is
$h_{e_1,e_2}:=(p+1-e_1-e_2)/\gcd(p-1, e_1+e_2-2).$
\end{itemize}
In all three cases, the tail cover is unique with the given ramification
when considered as a mere cover.
\end{lem}
\begin{proof}
Let $2\leq e\leq p-1$ be an integer. We define the primitive
tail cover $\varphi_e$ as the Galois closure of the degree-$p$ cover
$\bar{\varphi}_e:\bar{T}_e:={\mathbb P}^1\to {\mathbb P}^1$ given by
\begin{equation}\label{eq:tail1}
y^p+y^e=x, \qquad (x, y)\mapsto x.
\end{equation}
One easily checks that this is the unique degree-$p$ cover
between projective lines with one wild branch point and the required
tame ramification.
The decomposition group $G_e$ of $T_e$ is contained in $S_p$. We note
that the normalizer in $S_p$ of a Sylow $p$-subgroup has trivial
center. Therefore the inertia group $I$ of a wild ramification point
of $\varphi_e$ is contained in ${\mathbb F}_p\rtimes_\chi {\mathbb F}_p^\ast$, where
$\chi:{\mathbb F}_p^\ast\to {\mathbb F}_p^\ast$ is an injective character. Therefore
it follows from \cite{b-w5}, Proposition 2.2.(i) that $\gcd(h_e,
m_e)=1$. The statement on the wild ramification follows now directly
from the Riemann--Hurwitz formula (as in \cite{we1}, Lemma
4.10). Suppose that $e$ is odd. Then $m_e=(p-1)/\gcd(p-1, e-1)$
divides $(p-1)/2$. Therefore in this case both the tame and the wild
ramification groups are contained in $A_p$. This implies that the
Galois group $G_e$ of $\varphi_e$ is a subgroup of $A_p$.
To prove (a), we suppose that $e\neq p-1$. We show that the Galois
group $G_e$ of $\varphi_e$ is $A_p$ or $S_p$. Suppose that this is not the
case. Proposition \ref{prop:group} implies that
$e=p-2=2^r-1$. Moreover, $G_e$ is a subgroup of $\operatorname{P\Gamma L}_2(2^r)\simeq
\operatorname{PSL}_2(2^r)\rtimes {\mathbb Z}/r{\mathbb Z}$. The normalizer in $\operatorname{P\Gamma L}_2(2^r)$ of a
Sylow $p$-subgroup $I$ is ${\mathbb Z}/p{\mathbb Z}\rtimes {\mathbb Z}/2r{\mathbb Z}$. The
computation of the wild ramification shows that the inertia group
$I(\eta)$ of the wild ramification point $\eta$ is isomorphic to
${\mathbb Z}/p{\mathbb Z}\rtimes {\mathbb Z}/\frac{p-1}{2}{\mathbb Z}$. Therefore $\operatorname{P\Gamma L}_2(2^r)$
contains a subgroup isomorphic to $I(\eta)$ if and only if
$p=17=2^4+1$, in which case $I(\eta)=N_{\operatorname{P\Gamma L}_2(2^r)}(I)$. We
conclude that if $G_e\not\simeq S_p, A_p$ then $e=15$ and $p=17$. However,
in this last case one
may check using Magma that a suitable specialization of
(\ref{eq:tail1}) has Galois group $A_{17}$. As before, we conclude
that $G_e\simeq A_{17}$.
Now suppose that $e=p-1$. It is easy to see that the Galois closure
of $\bar{\varphi}_{p-1}$ is in this case the cover
$\varphi_{p-1}:{\mathbb P}^1\to {\mathbb P}^1$ obtained by dividing out ${\mathbb F}_p\rtimes
{\mathbb F}_p^\ast\subset \operatorname{PGL}_2(p) = \operatorname{Aut}({\mathbb P}^1)$. This proves (b).
Let $e_1, e_2$ be as in the statement of (c).
As in the proof of (a), we define $\varphi_{e_1,
e_2}$ as the Galois closure of a non-Galois cover
$\bar{\varphi}_{e_1, e_2}:\bar{T}_{e_1, e_2}\to {\mathbb P}^1$ of degree
$p$. The cover $\bar{\varphi}_{e_1, e_2}$, if it exists, is given by
an equation
\begin{equation}\label{eq:tail2}
F(y):=y^{e_1}(y-1)^{e_2}\tilde{F}(y)=x,\qquad (x, y)\mapsto x,
\end{equation}
where $\tilde{F}(y)=\sum_{i=0}^{p-e_1-e_2} c_i y^i$ has degree
$p-e_1-e_2$. We may assume that $c_{p-e_1-e_2}=1$. The condition that
$\bar{\varphi}_{e_1, e_2}$ has exactly three ramification points
$y=0,1,\infty$ yields the condition $F'(y)=\gamma
t^{e_1-1}(t-1)^{e_2-1}$. Therefore the coefficients of $\tilde{F}$
satisfy the recursion
\begin{equation}\label{eq:rec}
c_i=c_{i-1}\frac{e_1+e_2+i-1}{e_1+i}, \qquad i=1, \ldots, p-e_1-e_2.
\end{equation}
This implies that the $c_i$ are uniquely determined by
$c_{p-e_1-e_2}=1$. Conversely, it follows that the polynomial $F$
defined by these $c_i$ has the required tame ramification. The statement on the
wild ramification follows from the Riemann--Hurwitz formula, as in the
proof of (a).
\end{proof}
It remains to analyze the number of tail covers, and their automorphism
groups. Due to the nature of our argument, we will only need to carry
out this analysis for the tails of ramification type $e$.
From Lemma \ref{lem:tail1}, it follows already that the map
$\varphi_C:T_C\to {\mathbb P}^1$ is unique. However, part of the datum of a
tail cover is an isomorphism $\alpha:~\operatorname{Gal}(T_C, {\mathbb P}^1)\stackrel{\sim}{\to}
G_C$. For every $\tau\in \operatorname{Aut}(G_C)$, the tuple $(\varphi, \tau\circ
\alpha)$ also defines a tail cover, which is potentially
non-equivalent. Modification by $\tau$ leaves the cover unchanged as a
$G_C$-Galois cover if and only if $\tau \in \operatorname{Inn}(G_C)$. However, the weaker
notion of equivalence for tail covers implies that $\tau$ leaves the
cover unchanged as a tail cover if and only if $\tau$ can be described
as conjugation by an element of $N_{\operatorname{Aut}(T)}(G_C)$. Thus, the number of
distinct tail covers corresponding to a given mere cover is the order of
the cokernel of the map
$$ N_{\operatorname{Aut}(T_C)}(G_C) \to \operatorname{Aut}(G_C) $$
given by conjugation. Denote by
$\operatorname{Aut}_{G_C}(\varphi_C)$ the kernel of this map, or equivalently
the set of $G_C$-equivariant automorphisms of $T_C$. It follows
finally that the number of tail covers corresponding to $\varphi_C$
is
\begin{equation}\label{eq:number-tails}
\frac{|\operatorname{Aut}(G_C)||\operatorname{Aut}_{G_C}(\varphi_C)|}{|N_{\operatorname{Aut}(T_C)}(G_C)|}.
\end{equation}
Finally, denote by $\operatorname{Aut}_{G_C}^0(\varphi_C)\subset\operatorname{Aut}_{G_C}(\varphi_C)$ the
subset of automorphisms which fix the chosen ramification point $\eta$. We
now simultaneously compute
these automorphism groups and show that in the single-cycle case, we have
a unique tail cover.
\begin{lem}\label{lem:tail2} Let $2\leq e\leq p-1$ be an integer.
\begin{itemize}
\item[(a)] The group $\operatorname{Aut}_{G_e}(\varphi_e)$
(resp.\ $\operatorname{Aut}_{G_e}^0(\varphi_e)$) is cyclic of order $(p-e)/2$
(resp.\ $h_e$) if $e$ is odd and $p-e$ (resp.\ $h_e$) is $e$ is even.
\item[(b)] There is a unique primitive tail cover of type $e$.
\end{itemize}
\end{lem}
\begin{proof} First note that the definition of $\operatorname{Aut}_{G_e}(\varphi_e)$
implies that any element induces an automorphism of any intermediate
cover of $\varphi_e$, and in particular induces automorphisms of
$\bar{T}_e$ and ${\mathbb P}^1$.
Choose a primitive $(p-e)$th root of unity $\zeta\in
\bar{{\mathbb F}}_p$. Then $\mu(x, y)=(\zeta^ex, \zeta y)$ is an automorphism
of $\bar{T}_e$. One easily checks that $\mu$ generates the group of
automorphisms of $\bar{T}_e$ which induces automorphisms of ${\mathbb P}^1$
under $\varphi_e$, and that furthermore $T_e$ is Galois over
${\mathbb P}^1/\langle\mu\rangle$, so in particular every element of $\mu$ lifts
to an automorphism of $T_e$. Taking the quotient by the action of $\mu$,
we obtain a diagram
\begin{equation}\label{tail2eq}
\begin{CD}
\bar{T}_e @>>> \bar{T}'_e=\bar{T}_e/\langle\mu\rangle\\
@V{\bar{\varphi}_e}VV
@VV{\bar{\psi}_e}V\\
{\mathbb P}^1 @>>>{\mathbb P}^1/\langle\mu\rangle.
\end{CD}
\end{equation}
Since we know the ramification of the other three maps, one easily
computes that the tame ramification of $\bar{\psi}_e$ is
$e$-$(p-e)$. Let $\psi_e: T_e'\to {\mathbb P}^1$ be the Galois closure of
$\bar{\psi}_e$.
We now specialize to the case that $e$ is odd. Since
$G_e=A_p$ does not contain an element of cycle type $e$-$(p-e)$, it follows
that the Galois group $G'$ of $\psi_e$ is $S_p$.
Therefore it follows by degree considerations that the cover
$T_e \to T_e'$ is cyclic of degree $(p-e)/2$.
Denote by $Q$ the Galois group of the cover $T_e\to {\mathbb P}^1/\langle\mu\rangle$.
This is a group of order
$p!(p-e)/2$, which contains normal subgroups isomorphic to $A_p$ and
${\mathbb Z}/\frac{p-e}{2}{\mathbb Z}$, respectively.
It follows that $Q = {\mathbb Z}/\frac{p-e}{2}{\mathbb Z} \rtimes S_p$.
Note that $\operatorname{Aut}_{G_e}(\varphi_e)$ is necessarily a subgroup of
$Q$. In fact, it is precisely the subgroup of $Q$ which commutes
with every element of $A_p \subseteq Q$. One easily checks that the
semidirect product cannot be split, and that $\operatorname{Aut}_{G_e}(\varphi_e)$
is precisely the normal subgroup ${\mathbb Z}/\frac{p-e}{2}{\mathbb Z}$, that is the
Galois group of $T_e$ over $T'_e$.
To compute $\operatorname{Aut}^0_{G_e}(\varphi_e)$ we need to compute the
order of the inertia group of a wild ramification point of
$T_e$ in the map $T_e\to T'_e$.
Since a wild ramification point of $T_e'$
has inertia group of order $p(p-1)=pm_e\gcd(p-1, e-1)$, we know the
orders of the inertia groups of three of the four maps, and conclude
that $\operatorname{Aut}^0_{G_e}(\varphi_e)$ has order $h_e=(p-e)/\gcd(p-1, e-1)$.
This proves (a) in the case $e$ is odd.
For (b), we simply observe that since $Q \subset N_{\operatorname{Aut}(T_e)}(G_e)$,
we have
$$\frac{|\operatorname{Aut}(G_e)||\operatorname{Aut}_{G_e}(\varphi_e)|}{|N_{\operatorname{Aut}(T_e)}(G_e)|} \leq
\frac{p!\frac{p-e}{2}}{|Q|}=1,$$
so the tail cover is unique, as desired.
We now treat the case that $e$ is even. For (a), if $e<p-1$, the Galois
group of $\bar{\psi}_e$ is equal to the Galois group of $\bar{\varphi}_e$,
which is isomorphic to $S_p$. We conclude that the degree of
$T_e\to T'_e$ is $p-e$ in this case, and the group $Q$ defined as above
is a direct product ${\mathbb Z}/(p-e){\mathbb Z} \times S_p$.
Similarly to the case that $e$ is odd, we conclude that
$\operatorname{Aut}_{G_e}(\varphi_e)$ (resp.\ $\operatorname{Aut}_{G_e}^0(\varphi_e)$) is cyclic
of order $p-e$ (resp.\ $h_e$) in this case, as desired.
On the other hand, if $e=p-1$, we have
that $p-e=1$, hence $\mu$ is trivial, and we again conclude that (a) holds.
Finally, (b) is trivial: if $e<p-1$, the Galois group of $\varphi_e$ is
$S_p$ and $\operatorname{Aut}(S_p)=S_p$. Therefore there is a unique tail cover in this
case. The same conclusion holds in the case that $e=p-1$, since
$G_{p-1}\simeq {\mathbb F}_p\rtimes_\chi{\mathbb F}_p^\ast$ and $\operatorname{Aut}(G_{p-1})=G_{p-1}$.
The statement of the lemma follows.
\end{proof}
\begin{rem} In the case of $e_1$-$e_2$ tail covers, there may in fact
be more than one structure on a given mere cover. However, we will
not need to know this number for our argument.
\end{rem}
\section{Reduction of $3$-point covers}\label{sec:3pt}
In this section, we (almost) compute the number of $3$-point covers
with bad reduction for ramification types
$(p;e_1\text{-}e_2,e_3,e_4)$. More precisely, we compute this number
in the case that not both $e_3$ and $e_1+e_2$ are even. In the remaining
case, we only compute this number up to a factor $2$, which is good enough
for our purposes. Although we restrict to types of the above form,
our strategy applies somewhat more generally.
The results of this section rely on the results of Wewers \cite{we1},
who gives a precise formula for the number of lifts of a given special
$G$-map (Section \ref{sec:stable}) in the $3$-point case.
We fix a type $\tau=(p;e_1\text{-}e_2,e_3,e_4)$ satisfying the
genus-$0$ condition $\sum_i e_i=2p+2$. We allow $e_3$ or $e_4$ to be
$p$, although this is not the case that ultimately interests us; see
below for an explanation. We do however assume throughout that we are
not in the exceptional case $\tau=(5;2\text{-}2,4,4)$. According to
Lemma \ref{lem:tail1}, we may fix a set of primitive tail covers
$\bar{g}_i$ of type $C_i$, for $i$ such that $C_i\neq p$. Moreover,
by Proposition \ref{prop:dd} we have a (unique) deformation datum,
so we know there exists at least one special $G$-map $\bar{g}$ of type
$\tau$. Lemma \ref{lem:tail2} implies moreover that the number of special
$G$-maps is equal to the number of $e_1$-$e_2$ tail covers.
Wewers (\cite{we1}, Theorem 3) shows that there exists a $G$-Galois cover
$g:Y\to {\mathbb P}^1$ in characteristic zero with bad reduction to
characteristic $p$, and more specifically with stable reduction equal to
the given special $G$-map $\bar{g}$.
Moreover, Wewers gives a formula for the number
$\tilde{L}(\bar{g})$ for lifts of the given special $G$-map $\bar{g}$.
In order to state his formula, we need to introduce one more invariant.
Let $\operatorname{Aut}_G^0(\bar{g})$ be the group of $G$-equivariant automorphisms of
$\bar{Y}$ which induce the identity on the original component
$\bar{X}_0$.
Choose $\gamma\in \operatorname{Aut}_G^0(\bar{g})$, and consider the restriction of
$\gamma$ to the original component $\bar{X}_0$. Let $\bar{Y}_0$ be an
irreducible component of $\bar{Y}$ above $\bar{X}_0$ whose inertia
group is the fixed Sylow $p$-subgroup $I$ of $G$. As in (\ref{eq:G0}),
we write $G_0=I\rtimes_\chi H_0\subset {\mathbb F}_p\rtimes_\chi {\mathbb F}_p^\ast$
for the decomposition group of $\bar{Y}_0$. Wewers (\cite{we1}, proof
of Lemma 2.17) shows that $\gamma_0:=\gamma|_{\bar{Y}_0}$ centralizes
$H_0$ and normalizes $I$, i.e.\ $\gamma_0\in C_{N_G(I)}(H_0)$. Since
$\bar{Y}|_{\bar{X}_0}=\operatorname{Ind}_{G_0}^G\bar{Y}$ and $\gamma$ is
$G$-equivariant, it follows that the restriction of $\gamma$ to
$\bar{X}_0$ is uniquely determined by $\gamma_0$. We denote by $n'(\tau)$
the order of the subgroup consisting of those $\gamma_0\in C_{N_G(I)}(H_0)$
such that there exists a $\gamma\in \operatorname{Aut}_G^0(\bar{g})$ with
$\gamma|_{\bar{Y}_0}=\gamma_0$. Our notation is justified by Corollary
\ref{cor:nprime-defnd} below.
Wewers (\cite{we1}, Corollary 4.8) shows that
\begin{equation}\label{eq:pd}
|\tilde{L}(\bar{g})|=\frac{p-1}{n'(\tau)}\prod_{i\in
{\mathbb B}}\frac{h_{C_i}}{|\operatorname{Aut}_{G_{C_i}}^0(\bar{g}_{C_i})|}.
\end{equation}
The numbers are as defined in Section \ref{sec:stable}. (Note that the
group $\operatorname{Aut}_{G_{C_i}}^0(\bar{g}_{C_i})$ is defined differently from
the group $\operatorname{Aut}_G^0(\bar{g})$.)
To compute the number of curves with bad reduction, we need to
compute the number $n'(\tau)$ defined above. As explained by Wewers
(\cite[Lemma 2.17]{we1}), one may express the number $n'(\tau)$ in terms of
certain groups of automorphisms of the tail covers. However, there is a
mistake in the concrete description he gives of $\operatorname{Aut}_G^0(\bar{g})$ in
terms of the tails, therefore we do not use Wewers' description. For a
corrected version, we refer to the manuscript \cite{se6}.
The difficulty we face in using Wewers' formula directly is that
we do not know the Galois group $G_{e_1\text{-}e_2}$ of the $e_1$-$e_2$
tail. This prevents us from directly computing both the number of
$e_1$-$e_2$ tails, and the term $n'(\tau)$.
We avoid this problem by using the following trick.
We first consider covers of type $\tau^\ast=(p;
e_1\text{-}e_2, \varepsilon, p)$, with $\varepsilon=p+2-e_1-e_2$, which
all have bad reduction. This
observation lets us compute $n'(\tau^\ast)$ from Wewers' formula.
We then show that for covers of type $\tau=(p;
e_1\text{-}e_2,e_3,e_4)$, the number $n'(\tau)$ essentially only depends on
$e_1$ and $e_2$, allowing us to compute $n'(\tau)$ from $n'(\tau^\ast)$.
A problem with this method is that in the case that the Galois
groups of covers with type $\tau$ and $\tau^\ast$ are not equal, the
numbers $n'(\tau)$ and $n'(\tau^\ast)$ may
differ by a factor $2$. Therefore in this case, we are able to
determine the number of covers of type $\tau$ with bad reduction only
up to a factor $2$.
In Lemma \ref{lem:badtype}, we have counted non-Galois covers, but in
this section, we deal with Galois covers.
Let $G(\tau)$ be the Galois group of a cover of type $\tau$. This
group is well-defined and either $A_p$ or $S_p$, by Corollary
\ref{3pt-monodromy}. We write $\gamma(\tau)$ for the quotient of the
number of Galois covers of type $\tau$ by the Hurwitz number
$h(\tau)$. By Lemma \ref{Gallem}, it follows that $\gamma(\tau)$
is $2$ if $G$ is $A_p$ and $1$ if it is $S_p$. The number
$\gamma(\tau)$ will drop out from the formulas as soon as we
pass back to the non-Galois situation in Section \ref{sec:adm}.
We first compute the number
$n'(\tau^\ast)$. We note that by Corollary \ref{3pt-monodromy},
the Galois group $G(\tau^\ast)$ of a
cover of type $\tau^\ast$ is $A_p$ if $e_1+e_2$ is even and $S_p$
otherwise. In particular, we see that $G(\tau)=G(\tau^\ast)$ unless
$e_1+e_2$ and $e_3$ are both even. In this case we have that
$G(\tau)=S_p$ and $G(\tau^\ast)=A_p$. Recall from Lemma \ref{lem:tail2} that
there is a unique
tail cover for the single-cycle tails. We denote by $N_{e_1\text{-}e_2}$
the number of $e_1\text{-}e_2$ tails, and by $\operatorname{Aut}^0_{e_1\text{-}e_2}$
the group $\operatorname{Aut}^0_{G_{e_1\text{-}e_2}}(\bar{g}_{e_1\text{-}e_2})$ for
any tail cover $\bar{g}_{e_1\text{-}e_2}$ as in Lemma \ref{lem:tail1}.
Note that since $\bar{g}_{e_1\text{-}e_2}$ is unique as a mere cover,
and the definition of $\operatorname{Aut}^0_{G_{e_1\text{-}e_2}}(\bar{g}_{e_1\text{-}e_2})$
is independent of the $G$-structure, this notation is well-defined.
We similarly have from \eqref{eq:pd} that $|\tilde{L}(\bar{g})|$
depends only on $\tau$, so we write $\tilde{L}(\tau):=|\tilde{L}(\bar{g})|$
for any special $G$-map $\bar{g}$ of type $\tau$.
\begin{lem}\label{lem:tauast}
Let $\tau^\ast$ be as above. Then
\[
n'(\tau^\ast)= \frac{(1+\delta_{e_1,e_2})N_{e_1\text{-}e_2}(p-1)}
{\gcd(p-1, e_1+e_2-2)\gamma(\tau^\ast)|\operatorname{Aut}^0_{e_1\text{-}e_2}|}.
\]
Here $\delta_{e_1,e_2}$ is the Kronecker $\delta$.
\end{lem}
\begin{proof}
Lemma \ref{lem:badtype} implies that the Hurwitz number $h(\tau^\ast)$
equals $(p+1-e_1-e_2)/2$ if $e_1=e_2$ and $(p+1-e_1-e_2)$
otherwise. Since all covers of type $\tau^\ast$ have bad reduction,
$h(\tau^\ast)\gamma(\tau^\ast)$ is equal to
$N_{e_1\text{-}e_2} \cdot \tilde{L}(\tau^\ast)$. The statement of the
lemma follows by applying Lemmas \ref{lem:tail1}.(c), \ref{lem:tail2},
and (\ref{eq:pd}).
\end{proof}
We now analyze $n'$ in earnest. For convenience, for $i\in {\mathbb B}$ we also
introduce the notation $\widetilde{\operatorname{Aut}}_{G_i}(\bar{g}_i)$
for the group of $G$-equivariant automorphisms of the induced tail cover
$\operatorname{Ind}_{G_i}^G(\bar{g}_i)$. Recall also that $\xi_i$ is the node connecting
$\bar{X}_0$ to $\bar{X}_i$. We note that $n'$ may be analyzed tail by tail,
in the sense that given $\gamma_0 \in C_{N_G(I)}(H_0)$, we have that
$\gamma_0$ lifts to $\operatorname{Aut}^0(\bar{g})$ if and only if for each $i \in {\mathbb B}$,
there is some $\gamma_i \in \widetilde{\operatorname{Aut}}_{G_i}(\bar{g}_i)$ whose
action on $\bar{g}_i^{-1}(\xi_i)$ is compatible with $\gamma_0$.
The basic proposition underlying the behavior of $n'$ is then the following:
\begin{prop}\label{prop:nprime-desc} Suppose $G=S_p$ or $A_p$, and we have a
special $G$-map $\bar{g}:\bar{Y}\to\bar{X}$. Then:
\begin{itemize}
\item[(a)] For $i\in {\mathbb B}$, the
$G$-equivariant automorphisms of $\bar{g}^{-1}(\xi_i)$ form a cyclic group.
\item[(b)] Given an element $\gamma_0 \in C_{N_G(I)}(H_0)$ and $i \in {\mathbb B}$,
there exists $\gamma_i\in \widetilde{Aut}_{G_i}(\bar{g}_i)$
agreeing with the action of $\gamma_0$ on $\bar{g}^{-1}(\xi_i)$ if and only
if there exists $\gamma_i'\in \widetilde{Aut}_{G_i}(\bar{g}_i)$
having the same orbit length on $\bar{g}^{-1}(\xi_i)$ as $\gamma_0$ has.
\end{itemize}
\end{prop}
\begin{proof} For (a), if $\tilde{\xi}_i$ is a point above
$\xi_i$ lying on the chosen component $\bar{Y}_0$, one easily checks
that a $G$-equivariant automorphism $\gamma$ of $\bar{g}^{-1}(\xi_i)$ is
determined by where it sends $\tilde{\xi}_i$, which can in turn be
represented by an element $g \in G$ chosen so that
$g(\tilde{\xi}_i)=\gamma(\tilde{\xi}_i)$. Note that $\gamma \neq g$;
in fact, if $\gamma,\gamma'$ are determined by $g,g'$, the composition
law is that $\gamma \circ \gamma'$ corresponds to $g' g$.
Such a $g$ yields a choice of $\gamma$ if and only if we have
the equality of stabilizers $G_{\tilde{\xi}_i}=G_{g(\tilde{\xi}_i)}$.
Now, any $h \in G_{\tilde{\xi}_i}$ is necessarily in $G_0$,
and using that $I \subseteq G_{\tilde{\xi}_i}$, we find that we must have
$g I g^{-1} \subseteq G_0$. But $I$ contains the only $p$-cycles in
$G_0$, so we conclude $g\in N_G I$. However, since $I$ fixed
$\tilde{\xi}_i$, the choices of $G$ may be taken modulo $I$, so we
conclude that they lie in $N_G I/I$. Finally, since $G=S_p$ or $A_p$,
we have that $N_G I/I$ is cyclic, isomorphic to ${\mathbb Z}/(p-1){\mathbb Z}$ if
$G=S_p$ and to ${\mathbb Z}/(\frac{p-1}{2}){\mathbb Z}$ if $G=A_p$.
(b) then follows immediately, since the actions of both $\gamma_0$ and
$\gamma_i'$ on $\bar{g}^{-1}(\xi_i)$ lie in the same cyclic group; we
can take $\gamma_i$ to be an appropriate power of $\gamma_i'$.
\end{proof}
\begin{cor}\label{cor:nprime-defnd} For $\tau$ as above, $n'(\tau)$ is
well defined.
\end{cor}
\begin{proof} We know that $G=S_p$ or $A_p$, and we also know by
Proposition \ref{prop:dd} and Lemma \ref{lem:tail1} that the deformation
datum is uniquely determined, and so
are the tail covers, at least as mere covers. But the description of
$n'(\tau)$ given by Proposition \ref{prop:nprime-desc} is visibly
independent of the $G$-structure on the tail covers, so we obtain
the desired statement.
\end{proof}
We can now obtain the desired comparison of $n'(\tau)$ with $n'(\tau^\ast)$.
\begin{prop}\label{prop:n'}
Let $\tau=(p; e_1$-$e_2, e_3, e_4)$ be a type satisfying the genus-$0$
condition, and let $\tau^\ast$ be the corresponding modified
type.
Then if $G(\tau)=G(\tau^\ast)$ we have
$n'(\tau)=n'(\tau^\ast)$. Otherwise, $n'(\tau)\in \{2n'(\tau^\ast),
n'(\tau^\ast)\}$.
\end{prop}
\begin{proof}
Let $\gamma_0$ be a generator of $C_{N_G(I)}(H_0)$.
We ask which powers of $\gamma_0$ extend to an element of $\operatorname{Aut}_G^0(\bar{g})$,
and we analyze this question tail by tail. Fix a tail $\bar{X}_i$,
and suppose that it is a single-cycle tail of length $e:=e_i$. The crucial
assertion is that $\gamma_0$ itself (and hence all its powers) always
extends to $\bar{X}_i$.
First suppose that $e<p-1$ is even. Thus $G=G_i=S_p$, and
$\widetilde{\operatorname{Aut}}_G(\bar{g}_i)=\operatorname{Aut}_{G_i}(\bar{g}_i)$. Now, $\gamma_0$ acts
on the fiber of $\xi_i$ with orbit length $(p-1)/m_e=\gcd(p-1,p-e)$.
On the other hand, by Lemma \ref{lem:tail1} we have that
$h_e=(p-e)/\gcd(p-1,e-1)$. Lemma \ref{lem:tail2} implies
that if $\gamma_i \in \operatorname{Aut}_{G_i}(\bar{g}_i)$
is a generator, then the order of $\gamma_i$ is $p-e$, and also that
$\operatorname{Aut}^0_{G_i}(\bar{g}_i)$ has order $h_e$. We conclude that an orbit of
$\gamma_i$ has length $\gcd(p-1,e-1)=\gcd(p-1,p-e)$, and thus by Proposition
\ref{prop:nprime-desc} that $\gamma_0$ extends to $\bar{X}_i$, as claimed.
The next case is that $e$ is odd, and $G=A_p$.
This proceeds exactly as before, except that both orbits in question have
length $\gcd(p-1,p-e)/2$.
Now, suppose $e$ is odd, but $G=S_p$. Then the orbit length of
$\gamma_0$ is $\gcd(p-1,p-e)$.
We have $\widetilde{\operatorname{Aut}}_{G}(\bar{g}_i)$ equal to the $G$-equivariant
automorphisms of $\operatorname{Ind}_{A_p}^{S_p}(\bar{g}_i)$. These contain induced
copies of the $G$-equivariant automorphisms of $\bar{g}_i$, so in
particular we know we have elements of orbit length
$\gcd(p-1,e-1)/2$. However, in fact one also has a $G$-equivariant
automorphism exchanging the two copies of $\bar{g}_i$, and whose square
is the generator of the $A_p$-equivariant automorphisms of $\bar{g}_i$. One
may think of this as coming from the automorphism constructing in
Lemma \ref{lem:tail2} inducing the isomorphism
between the two different $A_p$-structures on the tail cover. We thus
have an element of $\widetilde{\operatorname{Aut}}_{G}(\bar{g}_i)$ of orbit length
$\gcd(p-1,e-1)$, and $\gamma_0$ extends to the tail in this case as well.
Finally, if $e=p-1$ then $m_i=p-1$ and thus $\gamma_0$ acts as the
identity on the fiber of $\xi_i$. The
claim is trivially satisfied in this case.
It follows that extending $\gamma_0$ to the $e$-tails imposes no
condition when $e<p$, and of course we do not have tails in the case
that $e=p$. Therefore the only non-trivial condition imposed in extending
$\gamma_0$ is the extension to the $e_1$-$e_2$-tail.
In the case that $G(\tau)=G(\tau^\ast)$ we conclude the desired statement
from Proposition \ref{prop:nprime-desc}, since the orbit lengths in
question are clearly the same in both cases. Suppose that $G(\tau)\neq
G(\tau^\ast)$. This happens if and only if both $e_1+e_2$ and $e_3$
are even. In this case we have that $G(\tau)=S_p$ and
$G(\tau^\ast)=A_p$. Here, we necessarily have that $e_1+e_2,e_3,e_4$ are all
even, so the only conditions imposed on either $n'(\tau)$ or $n'(\tau^\ast)$
come from the $e_1$-$e_2$ tail. Since the orbit of $\gamma_0$ is twice as
long in the case of $\tau$, the answers can differ by at most a factor of
$2$ in this case, as desired.
\end{proof}
Let $2\leq e_1\leq e_2\leq e_3\leq e_4<p$ be integers with $\sum_i
e_i=2p+2$ and $e_1+e_2\leq p$. The following corollary translates
Proposition \ref{prop:n'} into an estimate for the number of Galois
covers of type $\tau=(p; e_1$-$e_2, e_3, e_4)$ with bad
reduction. Theorem \ref{thm:3-hurwitz} is a special case.
\begin{cor}\label{cor:2cyclebad} Let $\tau=(p; e_1\text{-}e_2, e_3, e_4)$
with $\tau \neq (5;2\text{-}2,4,4)$. The number of mere covers of type $\tau$
with bad reduction to characteristic $p$ is equal to
\[
\begin{cases}
\delta(\tau)(p+1-e_1-e_2)&\text{ if }e_1\neq e_2,\\
\delta(\tau)(p+1-e_1-e_2)/2&\text{ if } e_1=e_2,
\end{cases}
\]
where $\delta(\tau)\in \{1,2\}$, and $\delta=1$ unless $e_1+e_2$ and $e_3$
are both even.
\end{cor}
\begin{proof}
We recall that the number of Galois covers of type $\tau$ with bad
reduction is equal to $N_{e_1\text{-}e_2} \cdot \tilde{L}(\tau^\ast)$.
It follows from Lemma
\ref{lem:tail1}.(c) and (\ref{eq:pd})
that this number is
\[
\begin{cases}
\displaystyle{
\frac{\gamma(\tau^\ast)n'(\tau^\ast)}{n'(\tau)}(p+1-e_1-e_2)}&
\text{ if }e_1\neq e_2,\\
\displaystyle{\frac{\gamma(\tau^\ast)n'(\tau^\ast)}{n'(\tau)}(p+1-e_1-e_2)/2}&
\text{ if }e_1= e_2.
\end{cases}
\]
The definition of the Galois factor $\gamma(\tau)$ implies that the
number of mere covers of type $\tau$ with bad reduction is
\[
\begin{cases}
\displaystyle{\frac{\gamma(\tau^\ast)}{\gamma(\tau)}
\frac{n'(\tau^\ast)}{n'(\tau)}(p+1-e_1-e_2)}& \text{ if }e_1\neq
e_2,\\ \displaystyle{\frac{\gamma(\tau^\ast)}{\gamma(\tau)}
\frac{n'(\tau^\ast)}{n'(\tau)}(p+1-e_1-e_2)/2}& \text{ if }e_1= e_2.
\end{cases}
\]
Proposition \ref{prop:n'} implies that $n'(\tau)/n'(\tau^\ast)\in
\{1,2\}$, and is equal to $1$ unless $e_1+e_2,e_3,e_4$ are all even.
Moreover, if $n'(\tau)\neq n'(\tau^\ast)$ then
$\gamma(\tau^\ast)/\gamma(\tau)=2$.
The statement of the corollary follows from this.
\end{proof}
\begin{rem} Similar to the proof of Corollary \ref{cor:2cyclebad}, one may
show that every genus-$0$ three-point cover of type $(p; e_1, e_2,
e_3)$ has bad reduction. We do not include this proof here, as a
proof of this result using linear series already occurs in
\cite{os7}, Theorem 4.2.
\end{rem}
\section{Reduction of admissible covers}\label{sec:adm}
In this section, we return to the case of non-Galois covers, and use the
results of Section \ref{sec:3pt} to compute the number of
``admissible covers with good reduction''. We start by defining what we
mean by this. As always, we fix a type $(p; e_1, e_2, e_3, e_4)$ with
$1<e_1\leq e_2\leq e_3\leq e_4<p$ satisfying the genus-$0$ condition
$\sum_i e_i=2p+2$. As in Section \ref{sec:char0}, we consider
admissible degenerations of type $(p;e_1, e_2,\ast,e_3, e_4)$, which
means that $Q_3=\lambda\equiv Q_4=\infty\pmod{p}$. Recall from
Section \ref{sec:char0} that in positive characteristic not every smooth
cover degenerates to an admissible cover, as a degeneration might
become inseparable. The number of admissible covers (even counted
with multiplicity) is still bounded by the number of smooth covers,
but equality need not hold.
\begin{defn}\label{def:phadm}
We define $h^{\rm\scriptstyle adm}_p(p; e_1, e_2,\ast,e_3, e_4)$ as the number of admissible
covers of type $(p; e_1,e_2, \ast, e_3,e_4)$, counted with multiplicity, over
an algebraically closed field of characteristic $p$.
\end{defn}
The following proposition is the main result of this
section.
\begin{prop}\label{prop:admbad}
The assumptions on the type $\tau=(p;e_1, e_2, e_3, e_4)$ are as above. Then
\[
h^{\rm\scriptstyle adm}_p(p; e_1, e_2,\ast, e_3, e_4)>h(p; e_1, e_2, e_3, e_4)-2p,
\]
and
\[
h^{\rm\scriptstyle adm}_p(p; e_1, e_2,\ast, e_3, e_4)=h(p; e_1, e_2, e_3, e_4)-p
\]
unless $e_1+e_2$ and $e_3$ are both even.
\end{prop}
\begin{proof}
We begin by noting that in the case $\tau=(5;2,2,4,4)$ corresponding to
the exceptional case of Corollary \ref{3pt-monodromy},
the assertion of the proposition is automatic since $h(5;2,2,4,4)=8<10$.
We may therefore assume that $\tau \neq (5;2,2,4,4)$.
We use the description of the admissible covers in characteristic
zero (Theorem \ref{degenerationlem}) and the results of Section
\ref{sec:3pt} to estimate the number of admissible covers with good
reduction to characteristic $p$, i.e.\ that remain separable.
We first consider the pure-cycle case, i.e.\ the case of Theorem
\ref{degenerationlem}.(a). Let $m$ be an integer satisfying the
conditions of {\it loc.\ cit.} We write ${f}_0:{V}_0\to {X}_0$ for the
corresponding admissible cover. Recall from Section \ref{sec:char0} that
$\bar{X}$ consists of two projective lines ${X}^1_0, {X}^2_0$
intersecting in one point. Choose an irreducible component ${Y}^i_0$
of ${Y}_0$ above ${X}^i_0$, and write ${f}^i_0:{Y}^i_0\to {X}^i_0$ for
the restriction. These are covers of type $(d_1; e_1, e_2, m)$ and
$(d_2; m, e_3, e_4)$ with $d_i\leq p$, respectively. The
admissible cover ${f}_0$ has good reduction to characteristic $p$ if
and only if both three-point covers ${f}^i_0$ have good
reduction.
It is shown in \cite{os7}, Theorem 4.2,
that a genus-$0$ three-point cover of type $(d;a,b,c)$ with $a,b,c<p$
has good reduction to characteristic $p$ if and only if its degree $d$
is strictly less than $p$.
Since the degree $d_2$ of the cover ${f}^2_0$ is always at least as
large as the other degree $d_1$, it is enough to calculate when
$d_2<p$. The Riemann--Hurwitz formula implies that
$d_2=(m+e_3+e_4-1)/2$. Therefore the condition $d_2<p$ is
equivalent to the inequality
$$e_3+e_4+m \leq 2p-1.$$
Since we assumed the existence of an admissible cover with $\rho$ an
$m$-cycle, it follows from
Theorem \ref{degenerationlem}.(a) that $m \leq
2d+1-e_3-e_4=2p+1-e_3-e_4$. We find that $d_2<p$ unless
$m=2p+1-e_3-e_4$. We also note that the lower bound for $m$ is always less
than or equal to the upper bound, which is $2p+1-e_3-e_4$. We thus conclude
that there are $2p+1-e_3-e_4$ admissible covers with bad reduction.
We now consider the case of an admissible cover with $\rho$ an
$e_1$-$e_2$-cycle (Theorem
\ref{degenerationlem}.(b)). Let $f_0:V_0 \to X_0$ be such an admissible
cover in characteristic $0$, as
above. In particular, the restriction $f_0^1$ (resp.\ $f_0^2$) has
type $(d_1; e_1, e_2, e_1\text{-}e_2)$ (resp.\ $(d_2; e_1\text{-}e_2, e_3,
e_4)$).
We write $g_0$ for the Galois closure of $f_0$, and $g_0^i$ for the
corresponding restrictions. Let $G^i$ be the Galois group of $g_0^i$.
The assumptions on the $e_i$ imply that $p$ does not divide the order
of Galois group of $g_0^1$, therefore $g_0^1$ has good reduction to
characteristic $p$. Moreover, the cover $g_0^1$ is uniquely
determined by the triple $(\rho^{-1}, g_3, g_4)$. If $e_1\neq
e_2$, the gluing is likewise uniquely determined, while if $e_1=e_2$ there
are exactly $2$ possibilities for the tuple $(g_1, g_2, g_3, g_4)$ for a
given triple $(\rho^{-1}, g_3, g_4)$. Therefore to count the number of
admissible covers with bad reduction in this case, it suffices to
consider the reduction behavior of the cover $g_0^2:Y_0^2\to
X_0^2$.
Corollary \ref{cor:2cyclebad} implies that whether or not $e_1$ equals
$e_2$, the number of admissible covers with bad reduction in the
$2$-cycle case is equal to $(p+1-e_1-e_2)$ unless $e_1+e_2$ and $e_3$ are
both even, and bounded from above
by $2(p+1-e_1-e_2)$ always.
We conclude using Theorem \ref{degenerationlem} that the total
number of admissible covers with bad reduction counted with
multiplicity is less than or equal to
\[
(2p+1-e_3-e_4)+2(p+1-e_1-e_2)=p+(p+1-e_1-e_2)<2p,
\]
and equal to
\[
(2p+1-e_3-e_4)+(p+1-e_1-e_2)=p
\]
unless $e_1+e_2$ and $e_3$ are both even.
The proposition follows.
\end{proof}
\begin{rem} Theorem 4.2 of \cite{os7} does not need the assumption $d=p$.
Therefore the proof of Proposition \ref{prop:admbad} in the
single-cycle case shows the following stronger result. Let $(d; e_1,
e_2, e_3, e_4)$ be a genus-$0$ type with $1<e_1\leq e_2\leq e_3\leq
e_4 < p$. Then the number of admissible covers with a single ramified
point over the node and bad reduction to characteristic $p$ is
$$(d-p+1)(d+p+1-e_3-e_4)$$
when either $d+1 \geq e_2+e_3$ or $d+1-e_1 < p$. Otherwise, all
admissible covers have bad reduction.
\end{rem}
\section{Proof of the main result}\label{sec:4pt}
In this section, we count the number of mere covers with
ramification type $(p;e_1, e_2, e_3, e_4)$ and bad reduction in the
case that the branch points are generic. Equivalently, we compute the
$p$-Hurwitz number $h_p(p;e_1, e_2, e_3, e_4)$.
Suppose that $r=4$ and fix a genus-$0$ type $\tau=(p; e_1, e_2, e_3,
e_4)$ with $2\leq e_1\leq e_2\leq e_3\leq e_4<p$.
We let $g:Y\to X= {\mathbb P}^1_K$ be a
Galois cover of type $\tau$ defined over a local field $K$ as in
Section \ref{sec:stable}, such that $(X; Q_i)$ is the generic $r$-marked
curve of genus $0$. It is no restriction to suppose that $Q_1=0,
Q_2=1, Q_3=\lambda, Q_4=\infty$, where $\lambda$ is transcendental
over ${\mathbb Q}_p$. We suppose that $g$ has bad reduction to characteristic
$p$, and denote by $\bar{g}:\bar{Y}\to \bar{X}$ the stable
reduction. We have seen in Section \ref{sec:stable} that we may associate
with $\bar{g}$ a set of primitive tail covers $( \bar{g}_i)$ and a
deformation datum $(\bar{Z}_0, \omega)$. The primitive tail
covers $\bar{g}_i$ for $i\in{\mathbb B}=\{1,2,3,4\}$ are uniquely
determined by the $e_i$ (Lemma \ref{lem:tail1}).
The following proposition shows that the number of covers with bad
reduction is divisible by $p$ in the case that the branch point are
generic.
\begin{prop} \label{prop:baddeg}
Suppose that $(X={\mathbb P}^1_K; Q_i)$ is the generic $r=4$-marked curve of
genus zero. Then the number of mere covers of $X$ of ramification type
$(p;e_1, e_2, e_3, e_4)$ with bad reduction is nonzero and divisible by $p$.
\end{prop}
\begin{proof}
Since the number of Galois covers and the number of mere covers
differ by a prime-to-$p$ factor, it suffices to prove the proposition
for Galois covers. The existence portion of the proposition is proved
in \cite{bo4}, Proposition 2.4.1, and the divisibility by $p$ in Lemma
3.4.1 of {\it loc.\ cit.}\ (in a more general setting). We briefly
sketch the proof, which is easier in our case due to the simple
structure of the stable reduction (Lemma \ref{lem:stablered}). The
idea of the proof is inspired by a result of \cite{we3}, Section~3.
We begin by observing that away from the wild branch point
$\xi_i$, the primitive tail cover $\bar{g}_i$ is tamely
ramified. Therefore we can lift this cover of affine curves to
characteristic zero.
Let ${\mathcal X}_{0}={\mathbb P}^1_R$ be equipped with $4$ sections $Q_1=0, Q_2=1,
Q_3=\lambda, Q_4=\infty$, where $\lambda\in R$ is transcendental over
${\mathbb Z}_p$. Then (\ref{eq:Kummer}) defines
an $m$-cyclic cover ${\mathcal Z}_0\to{\mathcal X}_0$. We write $Z\to X$ for its generic
fiber. Proposition \ref{prop:dd} implies the existence of a
deformation datum $(\bar{Z}_0, \omega)$. Associated with the deformation
datum is a character $\chi:{\mathbb Z}/m{\mathbb Z}\to {\mathbb F}_p^\times$ defined by
$\chi(\beta)=\beta^\ast z/z\pmod{z}$. The differential form $\omega$
corresponds to a $p$-torsion point $P_0\in J(\bar{Z}_0)[p]_\chi$ on
the Jacobian of $\bar{Z}_0$. See for example \cite{se7}
(Here we use that the conjugacy classes $C_i$ are
conjugacy classes of prime-to-$p$ elements. This implies that the
differential form $\omega$ is holomorphic.)
Since $\sum_{i=1}^4 h_{i}=2m$ and the branch points are generic, we
have that $J(\bar{Z}_0)[p]_\chi\simeq {\mathbb Z}/p{\mathbb Z}\times {\boldsymbol
\mu}_p$ (\cite{bo6}, Proposition 2.9) After
enlarging the discretely valued field $K$, if necessary, we may
choose a $p$-torsion point $P\in J({\mathcal Z}_0\otimes_R K)[p]_\chi$ lifting
$P_0$. It corresponds to an \'etale $p$-cyclic cover $W\to Z$. The
cover $\psi:W\to X$ is Galois, with Galois group
$N:={\mathbb Z}/p{\mathbb Z}\rtimes_\chi{\mathbb Z}/m{\mathbb Z}$. It is easy to see that $\psi$ has
bad reduction, and that its deformation datum is $(\bar{Z}_0,
\omega)$.
By using formal patching (\cite{ra3} or \cite{we1}), one now
checks that there exists a map $g_R:{\mathcal Y}\to {\mathcal X}$ of stable curves over
$\operatorname{Spec}(R)$ whose generic fiber is a $G$-Galois cover of smooth curves,
and whose special fiber defines the given tails covers and the
deformation datum. Over a neighborhood of the original component $g_R$
is the induced cover $\operatorname{Ind}_N^G {\mathcal Z}_0\to {\mathcal X}_0$. Over the tails, the
cover $g_R$ is induced by the lift of the tail covers. The fact that
we can patch the tail covers with the cover over ${\mathcal X}_0$ follows
from the observation that $h_{i}<m_{i}$ (Lemma \ref{lem:tail1}), since
locally there a unique cover with this ramification (\cite{we1}, Lemma
2.12). This proves the existence statement.
The divisibility by $p$ now
follows from the observation that the set of lifts $P$ of the
$p$-torsion point $P_0\in J(\bar{Z}_0)[p]_\chi$ corresponding to the
deformation datum is a ${\boldsymbol \mu}_p$-torsor.
\end{proof}
We are now ready to prove our Theorem \ref{thm:main}, as well as
a slightly sharper version of Theorem \ref{thm:good-degen}.
\begin{thm}\label{thm:main2} Let $p$ be an odd prime and $k$ an algebraically
closed field of characteristic $p$. Suppose we are given
integers $2\leq e_1\leq e_2\leq e_3\leq e_4<p$. There exists a dense
open subset $U\subset {\mathbb P}^1_k$ such that for $\lambda\in U$ the
number of degree-$p$ covers with ramification type $(e_1, e_2, e_3, e_4)$
over the branch points $(0,1,\lambda,\infty)$ is given by the formula
$$h_p(e_1,\dots,e_4)=\min_i(e_i(p+1-e_i))-p.$$
Furthermore, unless both $e_1+e_2$ and $e_3$ are even, every such
cover has good degeneration under a degeneration of the base sending
$\lambda$ to $\infty$.
\end{thm}
\begin{proof} Proposition \ref{prop:baddeg} implies that the
number of covers with ramification type $(p;e_1, e_2, e_3, e_4)$ and
bad reduction is at least $p$. This implies that the generic Hurwitz
number $h_p(e_1,\ldots, e_4)$ is at most $\min_i(e_i(p+1-e_i))-p$.
Proposition \ref{prop:admbad} implies that the number of admissible
covers in characteristic $p$ strictly larger than
$\min_i(e_i(p+1-e_i))-2p$. Since the number of separable covers can
only decrease under specialization, we conclude that the generic
Hurwitz number equals $\min_i(e_i(p+1-e_i))-p$. This proves the
first statement, and the second follows immediately from Proposition
\ref{prop:admbad} in the situation that $e_1+e_2$ and $e_3$ are not
both even.
\end{proof}
\begin{rem} By using the results of \cite{bo4} one can prove a
stronger result than Theorem \ref{thm:main2}. We say that a
$\lambda\in {\mathbb P}^1_k\setminus\{0,1,\infty\}$ is {\em supersingular} if
it is a zero of the polynomial (\ref{eq:Hasseinv}) and {\em
ordinary} otherwise. Then the number of covers in characteristic
$p$ of type $(p; e_1, e_2, e_3, e_4)$ branched at $(0,1,\lambda,
\infty)$ is $h_p(p; e_1, e_2, e_3, e_4)$ if $\lambda$ is ordinary
and $h_p(p; e_1, e_2, e_3, e_4)-1$ if $\lambda$ is
supersingular. To prove this result, one needs to study the stable
reduction of the cover $\pi:\bar{{\mathcal H}}\to {\mathbb P}^1_\lambda$ of the
Hurwitz curve to the configuration space. We do not prove this
result here, as it would require too many technical details.
\end{rem}
\bibliographystyle{hamsplain}
| {'timestamp': '2009-06-09T19:52:34', 'yymm': '0906', 'arxiv_id': '0906.1793', 'language': 'en', 'url': 'https://arxiv.org/abs/0906.1793'} | ArXiv |
\section{Introduction and main results}
In this note we are interested in the existence versus non-existence of stable sub- and super-solutions of equations of the form
\begin{equation} \label{eq1}
-div( \omega_1(x) \nabla u ) = \omega_2(x) f(u) \qquad \mbox{in $ {\mathbb{R}}^N$,}
\end{equation} where $f(u)$ is one of the following non-linearities: $e^u$, $ u^p$ where $ p>1$ and $ -u^{-p}$ where $ p>0$. We assume that $ \omega_1(x)$ and $ \omega_2(x)$, which we call \emph{weights}, are smooth positive functions (we allow $ \omega_2$ to be zero at say a point) and which satisfy various growth conditions at $ \infty$. Recall that we say that a solution $ u $ of $ -\Delta u = f(u)$ in $ {\mathbb{R}}^N$ is stable provided
\[ \int f'(u) \psi^2 \le \int | \nabla \psi|^2, \qquad \forall \psi \in C_c^2,\] where $ C_c^2$ is the set of $ C^2$ functions defined on $ {\mathbb{R}}^N$ with compact support. Note that the stability of $u$ is just saying that the second variation at $u$ of the energy associated with the equation is non-negative. In our setting this becomes: We say a $C^2$ sub/super-solution $u$ of (\ref{eq1}) is \emph{stable} provided
\begin{equation} \label{stable}
\int \omega_2 f'(u) \psi^2 \le \int \omega_1 | \nabla \psi|^2 \qquad \forall \psi \in C_c^2.
\end{equation}
One should note that (\ref{eq1}) can be re-written as
\begin{equation*}
- \Delta u + \nabla \gamma(x) \cdot \nabla u ={ \omega_2}/{\omega_1}\ f(u) \qquad \text{ in $ \mathbb{R}^N$},
\end{equation*}
where
$\gamma = - \log( \omega_1)$ and on occasion we shall take this point of view.
\begin{remark} \label{triv} Note that if $ \omega_1$ has enough integrability then it is immediate that if $u$ is a stable solution of (\ref{eq1}) we have $ \int \omega_2 f'(u) =0 $ (provided $f$ is increasing). To see this let $ 0 \le \psi \le 1$ be supported in a ball of radius $2R$ centered at the origin ($B_{2R}$) with $ \psi =1$ on $ B_R$ and such that $ | \nabla \psi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$. Putting this $ \psi$ into $ (\ref{stable})$ one obtains
\[ \int_{B_R} \omega_2 f'(u) \le \frac{C}{R^2} \int_{R < |x| <2R} \omega_1,\] and so if the right hand side goes to zero as $ R \rightarrow \infty$ we have the desired result.
\end{remark}
The existence versus non-existence of stable solutions of $ -\Delta u = f(u)$ in $ {\mathbb{R}}^N$ or $ -\Delta u = g(x) f(u)$ in $ {\mathbb{R}}^N$ is now quite well understood, see \cite{dancer1, farina1, egg, zz, f2, f3, wei, ces, e1, e2}. We remark that some of these results are examining the case where $ \Delta $ is replaced with $ \Delta_p$ (the $p$-Laplacian) and also in many cases the authors are interested in finite Morse index solutions or solutions which are stable outside a compact set.
Much of the interest in these Liouville type theorems stems from the fact that the non-existence of a stable solution is related to the existence of a priori estimates for stable solutions of a related equation on a bounded domain.
In \cite{Ni} equations similar to $ -\Delta u = |x|^\alpha u^p$ where examined on the unit ball in $ {\mathbb{R}}^N$ with zero Dirichlet boundary conditions. There it was shown that for $ \alpha >0$ that one can obtain positive solutions for $ p $ supercritical with respect to Sobolev embedding and so one can view that the term $ |x|^\alpha$ is restoring some compactness. A similar feature happens for equations of the form
\[ -\Delta u = |x|^\alpha f(u) \qquad \mbox{in $ {\mathbb{R}}^N$};\] the value of $ \alpha$ can vastly alter the existence versus non-existence of a stable solution, see \cite{e1, ces, e2, zz, egg}.
We now come to our main results and for this we need to define a few quantities:
\begin{eqnarray*}
I_G&:=& R^{-4t-2} \int_{ R < |x|<2R} \frac{ \omega_1^{2t+1}}{\omega_2^{2t}}dx , \\
J_G&:=& R^{-2t-1} \int_{R < |x| <2R} \frac{| \nabla \omega_1|^{2t+1} }{\omega_2^{2t}} dx ,\\I_L&:=& R^\frac{-2(2t+p-1)}{p-1} \int_{R<|x|<2R }{ \left( \frac{w_1^{p+2t-1}}{w_2^{2t}} \right)^{\frac{1}{p-1} } } dx,\\ J_L&:= &R^{-\frac{p+2t-1}{p-1} } \int_{R<|x|<2R }{ \left( \frac{|\nabla w_1|^{p+2t-1}}{w_2^{2t}} \right)^{\frac{1}{p-1} } } dx,\\
I_M &:=& R^{-2\frac{p+2t+1}{p+1} } \int_{R<|x|<2R }{ \left( \frac{w_1^{p+2t+1}}{w_2^{2t}} \right)^{\frac{1}{p+1} } } \ dx, \\
J_M &:= & R^{-\frac{p+2t+1}{p+1} } \int_{R<|x|<2R }{ \left( \frac{|\nabla w_1|^{p+2t+1}}{w_2^{2t}} \right)^{\frac{1}{p+1} } } dx.
\end{eqnarray*}
The three equations we examine are
\[ -div( \omega_1 \nabla u ) = \omega_2 e^u \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad (G), \]
\[ -div( \omega_1 \nabla u ) = \omega_2 u^p \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad (L), \]
\[ -div( \omega_1 \nabla u ) = - \omega_2 u^{-p} \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad (M),\] and where we restrict $(L)$ to the case $ p>1$ and $(M)$ to $ p>0$. By solution we always mean a $C^2$ solution. We now come to our main results in terms of abstract $ \omega_1 $ and $ \omega_2$. We remark that our approach to non-existence of stable solutions is the approach due to Farina, see \cite{f2,f3,farina1}.
\begin{thm} \label{main_non_exist} \begin{enumerate}
\item There is no stable sub-solution of $(G)$ if $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$ for some $0<t<2$.
\item There is no positive stable sub-solution (super-solution) of $(L)$ if $ I_L,J_L \rightarrow 0$ as $ R \rightarrow \infty$ for some $p- \sqrt{p(p-1)} < t<p+\sqrt{p(p-1)} $ ($0<t<\frac{1}{2}$).
\item There is no positive stable super-solution of (M) if $ I_M,J_M \rightarrow 0$ as $ R \rightarrow \infty$ for some $0<t<p+\sqrt{p(p+1)}$.
\end{enumerate}
\end{thm} If we assume that $ \omega_1$ has some monotonicity we can do better. We will assume that the monotonicity conditions is satisfied for big $x$ but really all ones needs is for it to be satisfied on a suitable sequence of annuli.
\begin{thm} \label{mono} \begin{enumerate} \item There is no stable sub-solution of $(G)$ with $ \nabla \omega_1(x) \cdot x \le 0$ for big $x$ if $ I_G \rightarrow 0$ as $ R \rightarrow \infty$ for some $0<t<2$.
\item There is no positive stable sub-solution of $(L)$ provided $ I_L \rightarrow 0$ as $ R \rightarrow \infty$ for either:
\begin{itemize}
\item some $ 1 \le t < p + \sqrt{p(p-1)}$ and $ \nabla \omega_1(x) \cdot x \le 0$ for big $x$, or \\
\item some $ p - \sqrt{p(p-1)} < t \le 1$ and $ \nabla \omega_1(x) \cdot x \ge 0$ for big $ x$.
\end{itemize}
There is no positive super-solution of $(L)$ provided $ I_L \rightarrow 0$ as $ R \rightarrow \infty$ for some $ 0 < t < \frac{1}{2}$ and $ \nabla \omega_1(x) \cdot x \le 0$ for big $x$.
\item There is no positive stable super-solution of $(M)$ provided $ I_M \rightarrow 0$ as $ R \rightarrow \infty$ for some $0<t<p+\sqrt{p(p+1)}$.
\end{enumerate}
\end{thm}
\begin{cor} \label{thing} Suppose $ \omega_1 \le C \omega_2$ for big $ x$, $ \omega_2 \in L^\infty$, $ \nabla \omega_1(x) \cdot x \le 0$ for big $ x$.
\begin{enumerate} \item There is no stable sub-solution of $(G)$ if $ N \le 9$.
\item There is no positive stable sub-solution of $(L)$ if $$N<2+\frac{4}{p-1} \left( p+\sqrt{p(p-1)} \right).$$
\item There is no positive stable super-solution of $(M)$ if $$N<2+\frac{4}{p+1} \left( p+\sqrt{p(p+1)} \right).$$
\end{enumerate}
\end{cor}
If one takes $ \omega_1=\omega_2=1$ in the above corollary, the results obtained for $(G)$ and $(L)$, and for some values of $p$ in $(M)$, are optimal, see \cite{f2,f3,zz}.
We now drop all monotonicity conditions on $ \omega_1$.
\begin{cor} \label{po} Suppose $ \omega_1 \le C \omega_2$ for big $x$, $ \omega_2 \in L^\infty$, $ | \nabla \omega_1| \le C \omega_2$ for big $x$.
\begin{enumerate} \item There is no stable sub-solution of $(G)$ if $ N \le 4$.
\item There is no positive stable sub-solution of $(L)$ if $$N<1+\frac{2}{p-1} \left( p+\sqrt{p(p-1)} \right).$$
\item There is no positive super-solution of $(M)$ if $$N<1+\frac{2}{p+1} \left( p+\sqrt{p(p+1)} \right).$$
\end{enumerate}
\end{cor}
Some of the conditions on $ \omega_i$ in Corollary \ref{po} seem somewhat artificial. If we shift over to the advection equation (and we take $ \omega_1=\omega_2$ for simplicity)
\[ -\Delta u + \nabla \gamma \cdot \nabla u = f(u), \] the conditions on $ \gamma$ become: $ \gamma$ is bounded from below and has a bounded gradient.
In what follows we examine the case where $ \omega_1(x) = (|x|^2 +1)^\frac{\alpha}{2}$ and $ \omega_2(x)= g(x) (|x|^2 +1)^\frac{\beta}{2}$, where $ g(x) $ is positive except at say a point, smooth and where $ \lim_{|x| \rightarrow \infty} g(x) = C \in (0,\infty)$. For this class of weights we can essentially obtain optimal results.
\begin{thm} \label{alpha_beta} Take $ \omega_1 $ and $ \omega_2$ as above.
\begin{enumerate}
\item If $ N+ \alpha - 2 <0$ then there is no stable sub-solution for $(G)$, $(L)$ (here we require it to be positive) and in the case of $(M)$ there is no positive stable super-solution. This case is the trivial case, see Remark \ref{triv}. \\
\textbf{Assumption:} For the remaining cases we assume that $ N + \alpha -2 > 0$.
\item If $N+\alpha-2<4(\beta-\alpha+2)$ then there is no stable sub-solution for $ (G)$.
\item If $N+\alpha-2<\frac{ 2(\beta-\alpha+2) }{p-1} \left( p+\sqrt{p(p-1)} \right)$ then there is no positive stable sub-solution of $(L)$.
\item If $N+\alpha-2<\frac{2(\beta-\alpha+2) }{p+1} \left( p+\sqrt{p(p+1)} \right)$ then there is no positive stable super-solution of $(M)$.
\item Further more 2,3,4 are optimal in the sense if $ N + \alpha -2 > 0$ and the remaining inequality is not satisfied (and in addition we assume we don't have equality in the inequality) then we can find a suitable function $ g(x)$ which satisfies the above properties and a stable sub/super-solution $u$ for the appropriate equation.
\end{enumerate}
\end{thm}
\begin{remark} Many of the above results can be extended to the case of equality in either the $ N + \alpha - 2 \ge 0$ and also the other inequality which depends on the equation we are examining. We omit the details because one cannot prove the results in a unified way.
\end{remark}
In showing that an explicit solution is stable we will need the weighted Hardy inequality given in \cite{craig}.
\begin{lemma} \label{Har}
Suppose $ E>0$ is a smooth function. Then one has
\[ (\tau-\frac{1}{2})^2 \int E^{2\tau-2} | \nabla E|^2 \phi^2 + (\frac{1}{2}-\tau) \int (-\Delta E) E^{2\tau-1} \phi^2 \le \int E^{2\tau} | \nabla \phi|^2,\] for all $ \phi \in C_c^\infty({\mathbb{R}}^N)$ and $ \tau \in {\mathbb{R}}$.
\end{lemma} By picking an appropriate function $E$ this gives,
\begin{cor} \label{Hardy}
For all $ \phi \in C_c^\infty$ and $ t , \alpha \in {\mathbb{R}}$. We have
\begin{eqnarray*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2 &\ge& (t+\frac{\alpha}{2})^2 \int |x|^2 (1+|x|^2)^{-2+\frac{\alpha}{2}}\phi^2\\
&&+(t+\frac{\alpha}{2})\int (N-2(t+1) \frac{|x|^2}{1+|x|^2}) (1+|x|^2)^{-1+\frac{\alpha} {2}} \phi^2.
\end{eqnarray*}
\end{cor}
\section{Proof of main results}
\textbf{ Proof of Theorem \ref{main_non_exist}.} (1). Suppose $ u$ is a stable sub-solution of $(G)$ with $ I_G,J_G \rightarrow 0$ as $ R \rightarrow \infty$ and let $ 0 \le \phi \le 1$ denote a smooth compactly supported function. Put $ \psi:= e^{tu} \phi$ into (\ref{stable}), where $ 0 <t<2$, to arrive at
\begin{eqnarray*}
\int \omega_2 e^{(2t+1)u} \phi^2 &\le & t^2 \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \\
&& +\int \omega_1 e^{2tu}|\nabla \phi|^2 + 2 t \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi.
\end{eqnarray*} Now multiply $(G)$ by $ e^{2tu} \phi^2$ and integrate by parts to arrive at
\[ 2t \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \le \int \omega_2 e^{(2t+1) u} \phi^2 - 2 \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi,\]
and now if one equates like terms they arrive at
\begin{eqnarray} \label{start}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^2 & \le & \int \omega_1 e^{2tu} \left( | \nabla \phi |^2 - \frac{ \Delta \phi}{2} \right) dx \nonumber \\
&& - \frac{1}{2} \int e^{2tu} \phi \nabla \omega_1 \cdot \nabla \phi.
\end{eqnarray} Now substitute $ \phi^m$ into this inequality for $ \phi$ where $ m $ is a big integer to obtain
\begin{eqnarray} \label{start_1}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m} & \le & C_m \int \omega_1 e^{2tu} \phi^{2m-2} \left( | \nabla \phi |^2 + \phi |\Delta \phi| \right) dx \nonumber \\
&& - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi
\end{eqnarray} where $ C_m$ and $ D_m$ are positive constants just depending on $m$. We now estimate the terms on the right but we mention that when ones assume the appropriate monotonicity on $ \omega_1$ it is the last integral on the right which one is able to drop.
\begin{eqnarray*}
\int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2 & = & \int \omega_2^\frac{2t}{2t+1} e^{2tu} \phi^{2m-2} \frac{ \omega_1 }{\omega_2^\frac{2t}{2t+1}} | \nabla \phi|^2 \\
& \le & \left( \int \omega_2 e^{(2t+1) u} \phi^{(2m-2) \frac{(2t+1)}{2t}} dx \right)^\frac{2t}{2t+1}\\ &&\left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1}.
\end{eqnarray*}
Now, for fixed $ 0 <t<2$ we can take $ m $ big enough so $ (2m-2) \frac{(2t+1)}{2t} \ge 2m $ and since $ 0 \le \phi \le 1$ this allows us to replace the power on $ \phi$ in the first term on the right with $2m$ and hence we obtain
\begin{equation} \label{three}
\int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2 \le \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} \left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1}.
\end{equation} We now take the test functions $ \phi$ to be such that $ 0 \le \phi \le 1$ with $ \phi $ supported in the ball $ B_{2R}$ with $ \phi = 1 $ on $ B_R$ and $ | \nabla \phi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$. Putting this choice of $ \phi$ we obtain
\begin{equation} \label{four}
\int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi |^2 \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}.
\end{equation} One similarly shows that
\[ \int \omega_1 e^{2tu} \phi^{2m-1} | \Delta \phi| \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}.\]
So, combining the results we obtain
\begin{eqnarray} \label{last} \nonumber \frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m} &\le& C_m \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}\\
&&- D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi.
\end{eqnarray}
We now estimate this last term. A similar argument using H\"{o}lder's inequality shows that
\[ \int e^{2tu} \phi^{2m-1} | \nabla \omega_1| | \nabla \phi| \le \left( \int \omega_2 \phi^{2m} e^{(2t+1) u} dx \right)^\frac{2t}{2t+1} J_G^\frac{1}{2t+1}. \] Combining the results gives that
\begin{equation} \label{last}
(2-t) \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{1}{2t+1} \le I_G^\frac{1}{2t+1} + J_G^\frac{1}{2t+1},
\end{equation} and now we send $ R \rightarrow \infty$ and use the fact that $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$ to see that
\[ \int \omega_2 e^{(2t+1) u} =0, \] which is clearly a contradiction. Hence there is no stable sub-solution of $(G)$.
(2). Suppose that $u >0$ is a stable sub-solution (super-solution) of $(L)$. Then a similar calculation as in (1) shows that for $ p - \sqrt{p(p-1)} <t < p + \sqrt{p(p-1)}$, $( 0 <t<\frac{1}{2})$ one has
\begin{eqnarray} \label{shit}
(p -\frac{t^2}{2t-1} )\int \omega_2 u^{2t+p-1} \phi^{2m} & \le & D_m \int \omega_1 u^{2t} \phi^{2(m-1)} (|\nabla\phi|^2 +\phi |\Delta \phi |) \nonumber \\
&& +C_m \frac{(1-t)}{2(2t-1)} \int u^{2t} \phi^{2m-1}\nabla \omega_1 \cdot \nabla \phi.
\end{eqnarray} One now applies H\"{o}lder's argument as in (1) but the terms $ I_L$ and $J_L$ will appear on the right hand side of the resulting
equation. This shift from a sub-solution to a super-solution depending on whether $ t >\frac{1}{2}$ or $ t < \frac{1}{2}$ is a result from the sign change of $ 2t-1$ at $ t = \frac{1}{2}$. We leave the details for the reader.
(3). This case is also similar to (1) and (2).
\hfill $ \Box$
\textbf{Proof of Theorem \ref{mono}.} (1). Again we suppose there is a stable sub-solution $u$ of $(G)$. Our starting point is (\ref{start_1}) and we wish to be able to drop the term
\[ - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi, \] from (\ref{start_1}). We can choose $ \phi$ as in the proof of Theorem \ref{main_non_exist} but also such that $ \nabla \phi(x) = - C(x) x$ where $ C(x) \ge 0$. So if we assume that $ \nabla \omega_1 \cdot x \le 0$ for big $x$ then we see that this last term is non-positive and hence we can drop the term. The the proof is as before but now we only require that $ \lim_{R \rightarrow \infty} I_G=0$.
(2). Suppose that $ u >0$ is a stable sub-solution of $(L)$ and so (\ref{shit}) holds for all $ p - \sqrt{p(p-1)} <t< p + \sqrt{p(p-1)}$. Now we wish to use monotonicity to drop the term from (\ref{shit}) involving the term $ \nabla \omega_1 \cdot \nabla \phi$. $ \phi$ is chosen the same as in (1) but here one notes that the co-efficient for this term changes sign at $ t=1$ and hence by restriction $t$ to the appropriate side of 1 (along with the above condition on $t$ and $\omega_1$) we can drop the last term depending on which monotonicity we have and hence to obtain a contraction we only require that $ \lim_{R \rightarrow \infty} I_L =0$. The result for the non-existence of a stable super-solution is similar be here one restricts $ 0 < t < \frac{1}{2}$.
(3). The proof here is similar to (1) and (2) and we omit the details.
\hfill $\Box$
\textbf{Proof of Corollary \ref{thing}.} We suppose that $ \omega_1 \le C \omega_2$ for big $ x$, $ \omega_2 \in L^\infty$, $ \nabla \omega_1(x) \cdot x \le 0$ for big $ x$. \\
(1). Since $ \nabla \omega_1 \cdot x \le 0$ for big $x$ we can apply Theorem \ref{mono} to show the non-existence of a stable solution to $(G)$. Note that with the above assumptions on $ \omega_i$ we have that
\[ I_G \le \frac{C R^N}{R^{4t+2}}.\] For $ N \le 9$ we can take $ 0 <t<2$ but close enough to $2$ so the right hand side goes to zero as $ R \rightarrow \infty$.
Both (2) and (3) also follow directly from applying Theorem \ref{mono}. Note that one can say more about (2) by taking the multiple cases as listed in Theorem \ref{mono} but we have choice to leave this to the reader.
\hfill $ \Box$
\textbf{Proof of Corollary \ref{po}.} Since we have no monotonicity conditions now we will need both $I$ and $J$ to go to zero to show the non-existence of a stable solution. Again the results are obtained immediately by applying Theorem \ref{main_non_exist} and we prefer to omit the details.
\hfill $\Box$
\textbf{Proof of Theorem \ref{alpha_beta}.} (1). If $ N + \alpha -2 <0$ then using Remark \ref{triv} one easily sees there is no stable sub-solution of $(G)$ and $(L)$ (positive for $(L)$) or a positive stable super-solution of $(M)$. So we now assume that $ N + \alpha -2 > 0$. Note that the monotonicity of $ \omega_1$ changes when $ \alpha $ changes sign and hence one would think that we need to consider separate cases if we hope to utilize the monotonicity results. But a computation shows that in fact $ I$ and $J$ are just multiples of each other in all three cases so it suffices to show, say, that $ \lim_{R \rightarrow \infty} I =0$. \\
(2). Note that for $ R >1$ one has
\begin{eqnarray*}
I_G & \le & \frac{C}{R^{4t+2}} \int_{R <|x| < 2R} |x|^{ \alpha (2t+1) - 2t \beta} \\
& \le & \frac{C}{R^{4t+2}} R^{N + \alpha (2t+1) - 2t \beta},
\end{eqnarray*} and so to show the non-existence we want to find some $ 0 <t<2$ such that
$ 4t+2 > N + \alpha(2t+1) - 2 t \beta$, which is equivalent to $ 2t ( \beta - \alpha +2) > (N + \alpha -2)$. Now recall that we are assuming that $ 0 < N + \alpha -2 < 4 ( \beta - \alpha +2) $ and hence we have the desired result by taking $ t <2$ but sufficiently close.
The proof of the non-existence results for
(3) and (4) are similar and we omit the details. \\
(5). We now assume that $N+\alpha-2>0$. In showing the existence of stable sub/super-solutions we need to consider $ \beta - \alpha + 2 <0$ and $ \beta - \alpha +2 >0$ separately.
\begin{itemize} \item $(\beta - \alpha + 2 <0)$ Here we take $ u(x)=0$ in the case of $(G)$ and $ u=1$ in the case of $(L)$ and $(M)$. In addition we take $ g(x)=\E$. It is clear that in all cases $u$ is the appropriate sub or super-solution. The only thing one needs to check is the stability. In all cases this reduces to trying to show that we have
\[ \sigma \int (1+|x|^2)^{\frac{\alpha}{2} -1} \phi^2 \le \int (1+|x|^2)^{\frac{\alpha}{2}} | \nabla\phi |^2,\] for all $ \phi \in C_c^\infty$ where $ \sigma $ is some small positive constant; its either $ \E$ or $ p \E$ depending on which equation were are examining.
To show this we use the result from Corollary \ref{Hardy} and we drop a few positive terms to arrive at
\begin{equation*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2\ge (t+\frac{\alpha}{2})\int \left (N-2(t+1) \frac{|x|^2}{1+|x|^2}\right) (1+|x|^2)^{-1+\frac{\alpha} {2}}
\end{equation*} which holds for all $ \phi \in C_c^\infty$ and $ t,\alpha \in {\mathbb{R}}$.
Now, since $N+\alpha-2>0$, we can choose $t$ such that $-\frac{\alpha}{2}<t<\frac{n-2}{2}$. So, the integrand function in the right hand side is positive and since for small enough $\sigma$ we have
\begin{equation*}
\sigma \le (t+\frac{\alpha}{2})(N-2(t+1) \frac{|x|^2}{1+|x|^2}) \ \ \ \text {for all} \ \ x\in \mathbb{R}^N
\end{equation*}
we get stability.
\item ($\beta-\alpha+2>0$) In the case of $(G)$ we take $u(x)=-\frac{\beta-\alpha+2}{2} \ln(1+|x|^2)$ and $g(x):= (\beta-\alpha+2)(N+(\alpha-2)\frac{|x|^2}{1+|x|^2})$. By a computation one sees that $u$ is a sub-solution of $(G)$ and hence we need now to only show the stability, which amounts to showing that
\begin{equation*}
\int \frac{g(x)\psi^2}{(1+|x|^{2 })^{-\frac{\alpha}{2}+1}}\le \int\frac{|\nabla\psi|^2}{ (1+|x|^2)^{-\frac{\alpha}{2}} },
\end{equation*} for all $ \psi \in C_c^\infty$. To show this we use Corollary \ref{Hardy}. So we need to choose an appropriate $t$ in $-\frac{\alpha}{2}\le t\le\frac{N-2}{2}$ such that for all $x\in {\mathbb{R}}^N$ we have
\begin{eqnarray*}
(\beta-\alpha+2)\left( N+ (\alpha-2)\frac{|x|^2}{1+|x|^2}\right) &\le& (t+\frac{\alpha}{2})^2 \frac{ |x|^2 }{(1+|x|^2}\\
&&+(t+\frac{\alpha}{2}) \left(N-2(t+1) \frac{|x|^2}{1+|x|^2}\right).
\end{eqnarray*}
With a simple calculation one sees we need just to have
\begin{eqnarray*}
(\beta-\alpha+2)&\le& (t+\frac{\alpha}{2}) \\
(\beta-\alpha+2) \left( N+ \alpha-2\right) & \le& (t+\frac{\alpha}{2}) \left(N-t-2+\frac{\alpha}{2}) \right).
\end{eqnarray*} If one takes $ t= \frac{N-2}{2}$ in the case where $ N \neq 2$ and $ t $ close to zero in the case for $ N=2$ one easily sees the above inequalities both hold, after considering all the constraints on $ \alpha,\beta$ and $N$.
We now consider the case of $(L)$. Here one takes $g(x):=\frac {\beta-\alpha+2}{p-1}( N+ (\alpha-2-\frac{\beta-\alpha+2}{p-1})
\frac{|x|^2}{1+|x|^2})$ and $ u(x)=(1+|x|^2)^{ -\frac {\beta-\alpha+2}{2(p-1)} }$. Using essentially the same approach as in $(G)$ one shows that $u$ is a stable sub-solution of $(L)$ with this choice of $g$. \\
For the case of $(M)$ we take $u(x)=(1+|x|^2)^{ \frac {\beta-\alpha+2}{2(p+1)} }$ and $g(x):=\frac {\beta-\alpha+2}{p+1}( N+ (\alpha-2+\frac{\beta-\alpha+2}{p+1})
\frac{|x|^2}{1+|x|^2})$.
\end{itemize}
\hfill $ \Box$
| {'timestamp': '2011-08-17T02:00:55', 'yymm': '1108', 'arxiv_id': '1108.3118', 'language': 'en', 'url': 'https://arxiv.org/abs/1108.3118'} | ArXiv |
\section{Introduction}
\label{sec:intro}
\begin{figure}
\centering
\begin{tabular}{c c c}
\begin{minipage}[c]{0.3\linewidth}
\includegraphics[width=\linewidth, height=1.4\linewidth]{res/img/example/2.jpg}
\end{minipage} &
\begin{minipage}[c]{0.3\linewidth}
\includegraphics[width=\linewidth, height=1.4\linewidth]{res/img/example/3.jpg}
\end{minipage} &
\begin{minipage}[c]{0.3\linewidth}
\includegraphics[width=\linewidth, height=1.4\linewidth]{res/img/example/5.jpg}
\end{minipage} \\
(a) & (b) & (c) \\
\multicolumn{3}{c}{\includegraphics[width=\linewidth]{res/img/imbalance.pdf}} \\
\multicolumn{3}{c}{(d)}\\
\end{tabular}
\caption{(a): $\textless$vase-sitting on-table$\textgreater$; (b): $\textless$man-sitting on-chair$\textgreater$; (c): $\textless$dog-sitting on-chair$\textgreater$. (a)(b)(c) have completely different visual appearances but are considered as the same relation class. (d): The long-tailed distribution of independent relation classes}
\label{fig:motivation}
\end{figure}
\begin{figure*}
\centering
\begin{tabular}{c c c}
\begin{minipage}[c]{0.36\linewidth}
\includegraphics[width=\linewidth]{res/img/intro_2.pdf}
\end{minipage} &
\begin{minipage}[c]{0.23\linewidth}
\includegraphics[width=\linewidth]{res/img/intro_0.pdf}
\end{minipage} &
\begin{minipage}[c]{0.36\linewidth}
\includegraphics[width=\linewidth]{res/img/intro_1.pdf}
\end{minipage} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{(a): The global knowledge graph of VG; (b): Unstructured output space in which the relation classifier is shared among all subject-object pairs; (c): Structured output space of HOSE-Net in which the relation classifier is context-specific.}
\label{fig:intro}
\end{figure*}
In recent years, visual recognition tasks for scene understanding has gained remarkable progress, particularly in object detection and instance segmentation. While accurate identification of objects is a critical part of visual recognition, higher-level scene understanding requires higher-level information of objects.
Scene graph generation aims to provide more comprehensive visual clues than individual object detectors by understanding object interactions. Such scene graphs serve as structural representations of images by describing objects as nodes (``subjects/objects") and their interactions as edges (``relation"), which benefit many high-level vision tasks such as image caption\cite{li2019know,yang2019auto,guo2019aligning}, visual question answering\cite{teney2017graph,peng2019cra} and image generation\cite{johnson2018image}.
In scene graph generation, we actually obtain a set of visual phases$\textless$ subject-relation-object $\textgreater$ and the locations of objects in the image.
The triples of each scene graph form a local knowledge graph of the image and the triples of the whole training set form a global knowledge graph of relationships as shown in Figure~\ref{fig:intro} (a).
It remains a challenging task because deep neural networks cannot directly predict structured data due to its continuous nature.
It's a common practice to divide the scene graphs into classifiable graph elements.
\cite{sadeghi2011recognition} divides them into visual phase classes.
However, it's infeasible due to the hyper-linear growth concerning the number of objects and relations.
A widely-adopted strategy is to divide them into independent object classes and relation classes\cite{lu2016visual}.
Most methods classify the objects separately and then apply local graph structures to learn contextual object representations for relation classification\cite{xu2017scene,qi2019attentive,chen2019knowledge}.
However, they ignore the fact that the output space should also be contextual and structure-aware and adopt an unstructured one as shown in Figure~\ref{fig:intro} (b).
Hence these methods suffer from drastic intra-class variations.
For example, given the relation ``sit on'', the visual contents vary from ``vase-sit on-table" to ``dog-sit on-chair" as shown in Figure~\ref{fig:motivation} (a)(b)(c).
On the other hand, the distribution of these independent relation classes is seriously unbalanced as shown in Figure~\ref{fig:motivation} (d).
To mitigate the issues mentioned above, we propose a novel higher order structure embedded network (HOSE-Net), which consists of a visual module, a structure-aware embedding-to-classifier (SEC) module and a hierarchical semantic aggregation(HSA) module.
The SEC module is designed to construct a contextual and structured output space.
First, since objects serve as contexts in relationships, SEC learns context embeddings which embeds the objects' behavior patterns as subjects or objects and transfers this knowledge among the classifiers it connects to based on the overall class structure.
It adopts a graph convolution network\cite{kipf2016semi} to propagate messages on the local graphs with the guidance of object co-occurrence\cite{mensink2014costa} statistics.
Second, SEC learns a mapping function to project the context embeddings to related relation classifiers.
This mapping function is shared among all contexts which implicitly encodes the statistical correlations among objects and relations and organize a global knowledge graph based output space shown in Figure~\ref{fig:intro} (c).
Since the unbalanced relation data are distributed into different subspaces, SEC can alleviate the long-tailed distribution and the intra-class variations.
However, even if the context-specific classifiers can share statistical strengths via the context embeddings, distributing the training samples into a large set of subspaces can still suffer from sparsity issues.
To address this problem, we are inspired by the thought that object-based contexts can be redundant or noisy since relations are often defined in more abstract contexts.
For example, ride in ``man-ride-horse'' and ``woman-ride-elephant'' can be summarized as ``people-ride-animal''.
Accordingly, we propose a hierarchical semantic aggregation (HSA) module to mine the latent higher order structures in the global knowledge graph.
HSA hierarchically clusters the graph nodes following the principle that, if two objects have similar behavior patterns in the relationships they involved, the contexts they create can be embedded together, which is designed to find a good strategy to redistribute the samples into a smaller set of subspaces.
An object semantic hierarchy is generated in the process even if HSA just uses the graph structures.
It's not hard to understand because a semantic hierarchy is based on the properties of objects which also very relevant to their behavior patterns in relationships.
In summary, the proposed Higher Order Structure Embedded Network(HOSE-Net) uses embedding methods to construct a structured output space.
By modeling the inter-dependencies among object labels and relation labels, the serious intra-class variations and the long-tailed distribution can be alleviated. Moreover, clustering methods are used to make the structured output space more scalable and generalized.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{res/img/overview.pdf}
\caption{The framework of our HOSE-Net. It consists of three modules: (1) a visual module which outputs detection results and prepare subject-object pairs for relation representation learning;
(2) a SEC module which embeds the object labels into context embeddings by message passing and maps them to classifiers;
(3) a HSA module which distill higher order structural information for context embedding learning.}
\label{fig:overview}
\end{figure*}
Our contributions are as follows:
\begin{enumerate}[(1)]
\item We propose to map the object-based contexts in relationships into a high-dimensional space to learn contextual and structure-aware relation classifiers via a novel structure-aware embedding-to-classifier module. This module can be integrated with other works focusing on visual feature learning.
\item We design a hierarchical semantic aggregation module to distill higher order structural information for learning a higher order structured output space.
\item We extensively evaluate our proposed HOSE-Net, which achieves new state-of-the-art performances on challenging Visual Genome \cite{krishna2017visual} and VRD~\cite{lu2016visual} benchmarks.
\end{enumerate}
\section{Related Work}
\textbf{Scene Graph Generation.}
Recently, the task of scene graph generation is proposed to understand the interactions between objects.
\cite{sadeghi2011recognition} decomposes the scene graphs into a set of visual phase classes and designs a detection model to directly detect them from the image.
Considering each visual phase as a distinct class would fail since the number of visual phase triples can be very large even with a moderate number of objects and relations.
An alternative strategy is to decompose the scene graphs into object classes and relation classes in which way the graph structures of the output data is completely collapsed.
Most of these methods focus on modeling the inter-dependencies of objects and relations in the visual representation learning.
\cite{dai2017detecting} embeds the statistical inference procedure into the deep neural network via structural learning.
\cite{xu2017scene} constructs bipartite sub-graphs of scene graphs and use RNNs to refine the visual features by iterative message passing.
\cite{chen2019knowledge,qi2019attentive,cui2018context} uses graph neural networks to learn contextual visual representation.
However, these methods still suffer from highly diverse appearances within each relation class because they all adopt a flat and independent relation classifiers.
In this paper, we argue that the structural information including the local and the global graph structures of the output data is vital for regularizing a semantic space.
\\
\textbf{Learning Correlated Classifiers With Knowledge Graph}.
Zero-shot learning(ZSL) models need to transfer the knowledge learned from the training classes to the unknown test classes.
A main research direction is to represent each class with learned vector representations.
In this way, the correlations between known classes and unknown classes can help to transfer the knowledge learned from the training classes to the unknown test classes by mapping the embeddings to visual classifiers.
Knowledge Graphs (KGs) effectively capture explicit relational knowledge about individual entities hence many methods\cite{wang2018zero,kampffmeyer2019rethinking,hascoet2019semantic,gao2019know,zhang2019tgg} use KGs to learn the class correlations.
In scene graph generation, the relation classes are correlated by object classes as in the knowledge graph and the structural information is vital for a well-defined output space.
We indirectly learn vector representations of the objects' role in relationships which are mapped to the visual relation classifiers via the knowledge graph structure.
\section{Approach}
\subsection{Overview}
We formally define a scene graph as $G = \left \{ B, O, R \right \}$. $O = \left \{ o_1, o_2, \dots, o_n \right \}$ is the object set and $o_i$ denotes the i-th object in image. $B = \left \{ b_1, b_2, \dots, b_n \right \}$ is the bounding box set and $b_i$ denotes the bounding box of $o_i$. $R = \left \{ r_{o_1 \rightarrow o_2}, r_{o_2 \rightarrow o_3}, \dots, r_{o_{(n-1)} \rightarrow o_n} \right \}$ is the edge set and $r_{o_i \rightarrow o_j}$ denotes the relation between subject $o_i$ and object $o_j$.
The probability distribution of the scene graph $\Pr(G | I)$ is formularized as:
\begin{equation}
\Pr(G | I) = \Pr(B|I)\Pr(O|B,I)\Pr(R|B,O,I)
\end{equation}
We follow the widely-adopted two-stage pipeline\cite{zellers2018neural} to generate scene graphs. The first stage is object detection including object localization ($\Pr(B|I)$) and object recognition ($Pr(O|B, I)$). The second stage is relation classification ($\Pr(R|B,O,I)$).
Our proposed HOSE-Net consists of a visual module, a SEC module and a HSA module.
Section ~\ref{sec:visrep} introduces the visual module.
The major component is an object detector,
which outputs $B$, $O$ and the region features $F=\left \{ f_1, f_2, \dots, f_n \right \}$.
Then a set of object pairs $\left \{ (f_s, f_t), (o_s, o_t), (b_s,b_t) \right \}$ are produced, where $ s \neq t;s, t = 1 ... n $.
The union box feature $f_u$ for each pair is extracted by a relation branch.
The spatial feature $f_{spt}$ for each pair is learned from $(b_s,b_t)$ by a spatial module.
Section ~\ref{sec:secm} introduces the structure-aware embedding-to-classifier(SEC) module.
First, we construct local graphs to transfer statistical information between context embeddings of $O$.
Then the context embeddings are mapped to a set of primitive classifiers.
The classifier for each relation representation is adaptively generated by concatenating the primitive classifiers according to the pair label $(o_s, o_t)$.
Section ~\ref{sec:hsa} introduces the hierarchical semantic aggregation(HSA) module.
Based on the resulting semantic hierarchy, HSA creates a context dictionary $\mathcal{D}$ to map $o_i$ to one-hot encoding $c_i \in \mathbb{R}^{K}$ where $K \in \left [ 1,N \right ] $ is the number of context embeddings and $N$ is the number of object classes.
The overall pipeline in shown in Figure~\ref{fig:overview}.
\subsection{Visual Representation}\label{sec:visrep}
\textbf{Object Detection}. In the first stage, the object detection is implemented by a Faster RCNN\cite{ren2015faster}. With the detection results, a set of subject-object region feature pairs $(f_s, f_t)$ with label information $(o_s, o_t)$ and coordinates of subject box $(x_s,y_s,w_s,h_s)$, object box $(x_t,y_t,w_t,h_t)$, union box$(x_u,y_u,w_u,h_u)$ is produced.
Then a separate relation branch uses three bottlenecks to refine the image feature and extract the union box feature $f_u$ of each subject-object pair by roi pooling.
While the Faster RCNN branch focuses on learning discriminative features for objects, the relation branch focuses on learning interactive parts of two objects.\\
\\
\textbf{Relation Representation}. Most existing methods explore the the visual representation learning for relations. To establish the connections between objects, they usually build graphs to associate the detected regions and use message passing frameworks to learn contextualized visual representations. Then the fusion features of the subjects and objects are projected to a set of independent relation labels by a softmax function.
Whether the relation classifiers are structured and contextualized has been little explored.
To verify the effectiveness of adopting a structured output space, we don't use a graph structure for learning the visual representations.
Given the triple region features from the detection module $(f_s, f_t, f_u)$, the visual representation of the relation is:
\begin{equation}
r_{st} = \Psi_{st}([f_u; f_s; f_t])
\end{equation}
where $[;]$ is the concatenation operation and $\Psi_{st}$ is a linear transformation. $[f_u, f_s, f_t] \in \mathbb{R}^{3d_f}$. \\
\\
\textbf{Spatial Representation}.
The relative positions of the subject boxes and the object boxes are also valuable spatical clues for recognizing the relations.
The normalized box coordinates $\widehat{b_i}$ are computed as $[\frac{x}{w_{img}},\frac{y}{h_{img}},\frac{x+w}{w_{img}},\frac{y+h}{h_{img}},\frac{wh}{w_{img} h_{img}}]$ where $w_{img}$ and $h_{img}$ are the width and height of the image.
The relative spatial feature $b_{st}$ is encoded as $[\frac{x_s-x_t}{w_t},\frac{y_s-y_t}{h_t},log\frac{w_s}{w_t},log\frac{h_s}{h_t}]$. The final spatial representation is the concatenation of the normalized features and the relative features of the subject and object boxes:
\begin{equation}
f_{spt} = \Psi_{spt}([\widehat{b_s},\widehat{b_t},b_{st}])
\end{equation}
where $\Psi_{spt}$ is a linear transformation, $[\widehat{b_s},\widehat{b_t},b_{st}] \in \mathbb{R}^{14}$.
\subsection{Structure-Aware Embedding-to-Classifier}\label{sec:secm}
Given the object label information $O = \left \{ o_1, o_2, \dots, o_n \right \}$, our proposed SEC module generates dynamic classifiers for relation representations according to the pair label $(o_s,o_t)$.
First, we embed the object labels into higher level context embeddings.
The one-hot context encodings of objects $\mathcal{C} = \left \{ c_1, c_2, \dots, c_n \right \},c_i \in \mathbb{R}^{K}$ are obtained through the context dictionary $\mathcal{D}$ which will be discussed in \ref{sec:hsa}.
The context embeddings $\mathcal{E} = \left \{ e_1, e_2, \dots, e_n \right \}, e_i \in \mathbb{R}^{d_e}$ are genereted as follows:
\begin{equation}
e_i = W_{e}c_i
\end{equation}
where $W_{e} \in \mathbb{R}^{d_e \times K}$ is a context embedding matrix to be learned.
Then the context embeddings $\mathcal{E}$ are fed into a graph convolution network to learn local contextual information based on object co-occurrences.
We model the co-occurrence pattern in the form of conditional probability, i.e., $\mathcal{P}_{ij}$, which denotes the probability of occurrence of the j-th object class when the i-th object class appears. We compute the co-occurrences of object pairs in the training set and get the matrix $\mathcal{T}\in \mathbb{R}^{N \times N}$, $N$ is the number of object classes. $\mathcal{T}_{ij}$ denotes the co-occurring times of label pairs. The conditional probability matrix $\mathcal{P}$ is computed by:
\begin{equation}
\mathcal{P}_{ij} = \frac{\mathcal{T}_{ij}}{\sum_{j}^{N}\mathcal{T}_{ij}}
\end{equation}
where $\sum_{j}^{N}\mathcal{T}_{ij}$ denotes the total number of i-th object class occurrences in the training set.
Then the adjacency matrix $A \in \mathbb{R}^{n \times n}$ of the local contextual graph is produced by:
\begin{equation}
A_{ij} = \mathcal{P}_{o_i, o_j}
\end{equation}
The update rule of each GCN layer is:
\begin{equation}
\mathcal{E} ^{l + 1} = f(\mathcal{E} ^{l}, A)
\end{equation}where $f$ is the graph convolution operation of \cite{kipf2016semi}.
The node output of the final GCN layer is the primitive classifiers $W_{prim} = \left \{ w_1, w_2, \dots, w_n \right \}, w_i \in \mathbb{R}^{\frac{d_{cls}}{2}}$ formulated as:
\begin{equation}
w_i = {e_i}^{l + 1} = \sigma(\sum_{j\in \mathbb{N}_{i}}A_{ij}U{e_j}^{l})
\end{equation}
where $U \in \mathbb{R}^{d_e \times \frac{d_{cls}}{2}}$ is the transformation matrix to be learned. $\mathbb{N}_{i}$ is the neighbor node set of $e_i$. $\sigma$ is the nonlinear function.
For each subject-object label pair $(o_s,o_t)$, the visual classifier $W_{st}$ is a composition of two primitive classifiers according to its context:
\begin{equation}
W_{st} = [w_{o_s};w_{o_t}] \in \mathbb{R}^{d_{cls}}
\end{equation}
where $[;]$ is the concatenation operation.
Apply the learned classifier to the relation representations to get the predicted scores:
\begin{equation}
\hat{y} = W_{st}[r_{st};f_{spt}]
\end{equation}
\subsection{Hierarchical Semantic Aggregation}\label{sec:hsa}
\begin{figure}
\centering
\begin{minipage}[c]{\linewidth}
\includegraphics[width=\linewidth]{res/img/graph_1.pdf}
\end{minipage}
(a)
\begin{minipage}[c]{\linewidth}
\includegraphics[width=\linewidth]{res/img/graph_2.pdf}
\end{minipage}
(b)
\caption{The connectivity subgraph of (street, sidewalk) is illustrated in two parts: (street, sidewalk) are objects (a) or subjects (b) in $\textless$subject-relation-object$\textgreater$. The blue edges are the connected path and
the yellow edges are the unconnected path.}
\label{fig:scene_graph_example}
\end{figure}
Even if the global knowledge graph exhibits rich, lower-order connectivity patterns captured at the level of objects and relations, new problems emerge if we create context embeddings for all object classes. When the number of classes $N$ increases, a lot of context-specific classifiers can't get sufficient training samples due to data sparsity and can not be scalable.
The motivation of HSA is, although relations exist among concrete objects, the objects actually have many similar higher level behavior patterns in the overall contexts.
And there exists higher order connectivity patterns on the class structure which are essential for understanding the object behaviors in relationships.
We design an clustering algorithm for mining the higher order structural information based on behavior patterns of objects.
The connectivity pattern with respect to two nodes $q_s, q_t$ of knowledge graph $KG$ is represented in a subgraph $SG$.
$SG$ includes two sets of connection nodes $L_s,L_o$ between $q_s, q_t$:
\begin{eqnarray}
q_i\in L_s \Leftrightarrow r_{q_i \rightarrow q_s} = r_{q_i \rightarrow q_t}\\
q_i\in L_o \Leftrightarrow r_{q_i \leftarrow q_s} = r_{q_i \leftarrow q_t}
\end{eqnarray}
where $r_{q_i \rightarrow q_j}$ denotes the relation between $q_i$ and $q_j$.
Figure~\ref{fig:scene_graph_example} illustrates an example, which visualize the common patterns between street and sidewalk: both of them are made of tile, can be covered in snow, a person can walk on them and have buildings nearby.
If the behavior patterns of $q_s$ and $q_t$ have a large overlap, they can be clustered into a higher-level node.
The similarity score between $q_s, q_t$ is defined as:
\begin{equation}
\begin{split}
f_{sim}(q_s, q_t) & = \frac{\left | L_s \right |}{d_{in}(q_s) + d_{in}(q_t) - \left | L_s \right |} \\
& + \frac{\left | L_o \right |}{d_{out}(q_s) + d_{out}(q_t) - \left | L_o \right |},
\end{split}
\end{equation}
where $d_{in}(q_i)/d_{out}(q_i)$ denotes the number of incoming/outgoing edges of node $q_i$, which represents the occurrence times of $q_i$ in all relationships as object/subject respectively. $\left | L_s \right|/\left | L_o \right |$ denotes the number of nodes in $L_s/L_o$, which represents the number of common behavior patterns of $q_s, q_t$ as object/subject.
This measure is fully based on the graph structure, not on the distributed representations of nodes from external knowledge graphs.
\\
\begin{algorithm}[tb]
\caption{}
\label{alg:optim}
\begin{algorithmic}[1]
\Require{$KG = \left \{ (q_1^0,q_2^0,...,q_N^0),(r_{q_1^0 \rightarrow q_2^0},...,r_{q_i^0 \rightarrow q_j^0},...) \right \}$,similarity measure function $f_{sim}$, cluster number $K$}
\For{\texttt{$i = 1,2,...,N$}}
\State{$\lambda_{i} = 1 $}
\For{\texttt{$j = 1,2,...,N$}}
\State{$Sim(i,j) = f_{sim}(q_i^0,q_j^0)$}
\State{$Sim(j,i) = Sim(i,j)$}
\EndFor
\EndFor
\State {Set current cluster number $num = N$}
\While{$num > K$}\Comment{Find the two most similar node cluster}
\State{ $q_{i}^{l_i}, q_{j}^{l_j} \gets SELECTMAX(Sim(i,j) / (\lambda_{i} + \lambda_{j}))$}
\State $q_{i}^{l_{ij}} \gets MERGE(q_{i}^{l_i},q_{j}^{l_j})$
\State $KG,Sim \gets UPDATE(KG,Sim)$
\State $\lambda_{i} \gets \lambda{i} + \lambda{j} + 1$
\State $REINDEX(\lambda)$
\State $num \gets num - 1$
\EndWhile
\State{$\mathcal{D} \gets GETDICT((q_1^0,q_2^0,...,q_N^0),(q_1^l,q_2^l,...,q_K^l))$}
\Ensure{$\mathcal{D}$}
\end{algorithmic}
\end{algorithm}
\noindent\textbf{Algorithm.} We use hierarchical agglomerative clustering to find the node clusters shown in Algorithm~\ref{alg:optim}.
At each iteration, we merge the two clusters which have the most similar behavior patterns and update the knowledge graph by replacing the two clustered nodes with a higher level node.
Since the given triples are incomplete and unbalanced, we introduce a penalty term $\lambda$ to avoid the objects which have frequent occurrences in annotated relationships dominating the clustering.
When the number of clusters reaches the given K, the algorithm stops iterating.
We encode the clustering results as a dictionary $\mathcal{D}$ to map the N objects to one-hot encodings of dimension $K$ hence the objects within the same cluster will have the same context embedding.
In this way, the output space is reorganized into a smaller one.
Even if the clusters are not reasonable for all relations, experiments show that the context embeddings can still learn the upside in a high-dimensional space.
\\
\begin{figure*}
\centering
\begin{tabular}{c c}
\begin{minipage}[c]{0.48\linewidth}
\includegraphics[height=\linewidth]{res/img/tree.pdf}
\end{minipage} &
\begin{minipage}[c]{0.52\linewidth}
\includegraphics[height=\linewidth]{res/img/tree1.pdf}
\end{minipage} \\
(a) & (b) \\
\end{tabular}
\caption{(a) The semantic hierarchy of VG. (b) The semantic hierarchy of VRD.}
\label{fig:optim_result}
\end{figure*}
\section{Experiments}
\begin{table*}
\centering
\begin{tabular}{l l | l | l l | l l | l l}
\hline
SEC & HSA & & \multicolumn{2}{c}{SGDET} \vline & \multicolumn{2}{c}{SGCLS} \vline & \multicolumn{2}{c}{PRDCLS}\\
& & Recall at & 50 & 100 & 50 & 100 & 50 & 100 \\
\hline
\xmark & \xmark & Baseline $(K=1)$ & 28.1 & 32.5 & 34.8 & 36.4 & 64.6 & 67.3\\
\cmark & \xmark & HOSE-Net $(K=150)$ & 28.6 & 33.1 & 36.2 & 37.3 & 66.5 & 69.0\\
\cmark & \cmark & HOSE-Net $(K=40)$ & \textbf{28.9} & \textbf{33.3} & \textbf{36.3} & \textbf{37.4} & \textbf{66.8} & \textbf{69.2}\\
\hline
\end{tabular}
\caption{Ablation study on the SEC module and HSA module.}
\label{tab:compair_m}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=\linewidth, height=0.45\linewidth]{res/img/ablation.pdf}
\caption{Ablation study on the clustering number. We draw the performance curves of SGCLS and PRDCLS on K = 1, 10, 20, 30, 40, 50, 70, 100, 130, 150.}
\label{fig:compair_m}
\end{figure}
\begin{table*}
\centering
\begin{tabular}{l | l l | l l | l l}
\hline
& \multicolumn{2}{c}{SGDET} \vline & \multicolumn{2}{c}{SGCLS} \vline & \multicolumn{2}{c}{PRDCLS}\\
Recall at & 50 & 100 & 50 & 100 & 50 & 100 \\
\hline
SEC ($K=150$) & 28.6 & 33.1 & 36.2 & 37.3 & 66.5 & 69.0\\
SEC + kmeans with word2vec embedding($K=40$) & 28.7 & 33.1 & 36.2 & 37.3 & 66.4 & 68.9\\
SEC + HSA ($K=40$) & \textbf{28.9} & \textbf{33.3} & \textbf{36.3} & \textbf{37.4} & \textbf{66.8} & \textbf{69.2}\\
\hline
\end{tabular}
\caption{ Comparison between the clustering in HSA module and kmeans with word2vec embedding clustering.}
\label{tab:kmeans}
\end{table*}
\begin{table*}
\centering
\begin{tabular}{l | l l l l l l | l l l l l l}
\hline
& \multicolumn{6}{c}{Graph Constraint} \vline & \multicolumn{6}{c}{No Graph Constraint}\\
& \multicolumn{2}{c}{SGDET} & \multicolumn{2}{c}{SGCLS} & \multicolumn{2}{c}{PRDCLS} \vline & \multicolumn{2}{c}{SGDET} & \multicolumn{2}{c}{SGCLS} & \multicolumn{2}{c}{PRDCLS}\\
Recall at & 50 & 100 & 50 & 100 & 50 & 100 & 50 & 100 & 50 & 100 & 50 & 100 \\
\hline
VRD~\cite{lu2016visual} & 0.3 & 0.5 & 11.8 & 14.1 & 27.9 & 35.0 & - & - & - & - & - & -\\
ISGG~\cite{xu2017scene} & 3.4 & 4.2 & 21.7 & 24.4 & 44.8 & 53.1 & - & - & - & - & - & -\\
MSDN~\cite{xu2017scene} & 7.0 & 9.1 & 27.6 & 29.9 & 53.2 & 57.9 & - & - & - & - & - & - \\
AsscEmbed~\cite{newell2017pixels} & 8.1 & 8.2 & 21.8 & 22.6 & 54.1 & 55.4 & 9.7 & 11.3 & 26.5 & 30.0 & 68.0 & 75.2\\
Message Passing+~\cite{zellers2018neural} & 20.7 & 24.5 & 34.6 & 35.4 & 59.3 & 61.3 & 22.0 & 27.4 & 43.4 & 47.2 & 75.2 & 83.6\\
Frequency~\cite{zellers2018neural} & 23.5 & 27.6 & 32.4 & 34.0 & 59.9 & 64.1 & 25.3 & 30.9 & 40.5 & 43.7 & 71.3 & 81.2\\
Frequency+Overlap~\cite{zellers2018neural} & 26.2 & 30.1 & 32.3 & 32.9 & 60.6 & 62.2 & 28.6 & 34.4 & 39.0 & 43.4 & 75.7 & 82.9\\
MotifNet-LeftRight~\cite{zellers2018neural} & 27.2 & 30.3 & 35.8 & 36.5 & 65.2 & 67.1 & 30.5 & 35.8 & \textbf{44.5} & 47.7 & 81.1 & 88.3\\
GraphRCNN~\cite{yang2018graph} & 11.4 & 13.7 & 29.6 & 31.6 & 54.2 & 59.1 & - & - & - & - & - & - \\
KERN~\cite{chen2019knowledge} & 27.1 & 29.8 & \textbf{36.7} & 37.4 & 65.8 & 67.6 & - & - & - & - & - & -\\
VCTREE~\cite{tang2019learning} & 27.7 & 31.1 & 37.9 & \textbf{38.6} & 66.2 & 67.9 & - & - & - & - & - & -\\
\hline
HOSE-Net ($K=40$) & \textbf{28.9} & \textbf{33.3} & 36.3 & 37.4 & \textbf{66.7} & \textbf{69.2} & \textbf{30.5} & \textbf{36.3} & 44.2 & \textbf{48.1} & \textbf{81.1} & \textbf{89.2}\\
\hline
\hline
RelDN$^\ast$~\cite{zhang2019graphical} & 28.3 & 32.7 & 36.8 & 36.8 & 68.4 & 68.4 & 30.4 & \textbf{36.7} & 48.9 & 50.8 & 93.8 & 97.8\\
HOSE-Net$^\ast$ ($K=40$) & \textbf{28.9} & \textbf{33.3} & \textbf{37.3} & \textbf{37.3} & \textbf{70.1} & \textbf{70.1} & \textbf{30.5} & 36.3 & \textbf{49.7} & \textbf{51.2} & \textbf{94.6} & \textbf{98.2}\\
\hline
\end{tabular}
\caption{Comparison with state-of-the-art methods on Visual Genome. HOSE-Net$^\ast$ uses the evaluation metric in ~\cite{zhang2019graphical}}
\label{tab:sota_vg}
\end{table*}
\subsection{Datasets}
\textbf{Visual Genome\cite{krishna2017visual}.} It is a large scale dataset with 75729 object classes and 40480 relation classes. There are several modified versions for scene graph generation. In this paper, we follow the same train/val splits in which the most frequent 150 objects and 50 relations are chosen.
We measure our method on VG in three tasks:
\begin{enumerate}
\item predicate classification (PRDCLS): Given the ground truth annotations of the object classes and bounding boxes, predict the relation type of each object pair.
\item Scene graph classification (SGCLS): Given the ground truth annotations of object bounding boxes, predict the object classes and the relation type of each object pair.
\item Scene graph detection (SGDET): Predict the bounding boxes, the object classes and the relation type of each object pair.
\end{enumerate}
We use Recall@50, Recall@100 as our evaluation metrics. Recall@x computes the fraction of relationship hits in the top x confident relationship predictions. The reason why precision and average precision (AP) are not proper metrics for this task is, only a fraction of relationships are annotated and they will penalize the right detection if it is not in the ground truth.
We report the Graph Constraint Recall@x following \cite{lu2016visual} which only involves the highest score relation prediction of each subject-object pair in the recall ranking. We also report the No Graph Constraint Recall@x following \cite{newell2017pixels} which involves all the 50 relation scores of each subject-object pair in the recall ranking. It allows multiple relations exist between objects.
\noindent\textbf{VRD\cite{lu2016visual}} contains 4000 training and 1000 test images including 100 object classes and 70 relations.
We follow \cite{yu2017visual} to measure our method on VRD in two tasks:
\begin{enumerate}
\item Phase detection: Predict the visual phase triplets $\textless$subject-relation-object$\textgreater$ and localize the union bounding boxes of each object pair.
\item Relationship detection: Predict the visual phase triplets $\textless$subject-relation-object$\textgreater$ and localize the bounding boxes of subjects and objects.
\end{enumerate}
We report Recall@50 and Recall@100 at involving 1 ,10 and 70 relation predictions per object pair in recall ranking as the evaluation metrics.
\begin{table*}
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}{l | l l l l l l | l l l l l l}
\hline
& \multicolumn{6}{c}{Relationship Detection} \vline & \multicolumn{6}{c}{Phrase Detection}\\
& \multicolumn{2}{c}{rel=1} & \multicolumn{2}{c}{rel=10} & \multicolumn{2}{c}{rel=70} \vline& \multicolumn{2}{c}{rel=1} & \multicolumn{2}{c}{rel=10} & \multicolumn{2}{c}{rel=70}\\
Recall at & 50 & 100 & 50 & 100 & 50 & 100 & 50 & 100 & 50 & 100 & 50 & 100\\
\hline
VTransE~\cite{zhang2017visual} & 19.4 & 22.4 & - & - & - & - & 14.1 & 15.2 & - & - & - & -\\
ViP-CNN~\cite{li2017vip} & 17.32 & 21.01 & - & - & - & - & 22.78 & 27.91 & - & - & - & -\\
VRL~\cite{liang2017deep} & 18.19 & 20.79 & - & - & - & - & 21.37 & 22.60 & - & - & - & -\\
KL distilation~\cite{yu2017visual} & 19.17 & 21.34 & \textbf{22.56} & \textbf{29.89} & \textbf{22.68} & \textbf{31.89} & 23.14 & 24.03 & 26.47 & 29.76 & 26.32 & 29.43\\
MF-URLN~\cite{zhan2019exploring} & 23.9 & 26.8 & - & - & - & - & 31.5 & 36.1 & - & - & - & -\\
Zoom-Net~\cite{yin2018zoom} & 18.92 & 21.41 & - & - & 21.37 & 27.30 & 24.82 & 28.09 & - & - & 29.05 & 37.34\\
CAI + SCA-M~\cite{yin2018zoom} & 19.54 & 22.39 & - & - & 22.34 & 28.52 & 25.21 & 28.89 & - & - & \textbf{29.64} & \textbf{38.39}\\
RelDN (ImageNet)~\cite{zhang2019graphical} & 19.82 & 22.96 & 21.52 & 26.38 & 21.52 & 26.38 & 26.37 & 31.42 & 28.24 & 35.44 & 28.24 & 35.44\\
\hline
HOSE-Net ($K=18$) & \textbf{20.46} & \textbf{23.57} & 22.13 & 27.36 & 22.13 & 27.36 & \textbf{27.04} & \textbf{31.71} & \textbf{28.89} & \textbf{36.16} & 28.89 & 36.16\\
\hline
\end{tabular}
}
\caption{Comparison with state-of-the-art methods on VRD.}
\label{tab:sota_vrd}
\end{table*}
\subsection{Implementation Details}
HOSE-Net adopts a two-stage pipeline. The object detector is Faster RCNN with a VGG backbone initialized by COCO pre-trained weights for Visual Genome and ImageNet pre-trained weights for VRD and then finetuned on the visual relationship datasets. The backbone weights are fixed.
For stable training, we add an unstructured relation classifier as a separate branch for joint training.
Considering the dataset scale and dataset quality, we adopt different training mechanisms for Visual Genome and VRD. In Visual Genome experiments, we set $lr = 0.001$ for the structured classifier and $lr = 0.01$ for the unstructured one. During testing, we evaluate the structured classifier. In VRD, the loss weight of the unstructured classifier is 0.7, and the structured one is 0.3.
During testing, the result is the weight sum of the two classifiers.
Since the statistical bias is a widely-adopted strategy in the two-stage pipeline, we train a bias vector and fuse the bias results with the visual module results during testing following \cite{zellers2018neural}.
The proposed framework is implemented by PyTorch. All experiments are conducted on servers with 8 NVIDIA Titan X GPUs with 12 GB memory.
The batch size at the training phase is set to 8. $d_f$ is set to 4096 and $d_e$ is set to 512.
\subsection{Ablation Study} Now we perform ablation studies to better examine the effectiveness of our framework.
\\
\textbf{Structured Output Space with Cluster Number K.} We perform an ablation study to validate the effectiveness of the SEC module which learns a structured output space and the HSA module which incorporates higher order structure into the output space with respect to the cluster number K.
$K = 1$ is our baseline model which uses the conventional unstructured relation classifiers.
$K = 150$ only employs SEC module to learn a low order structured output space.
In HSA module, K is a hyper parameter which can be a trade-off between the performance and the model complexity.
We know that all clustering algorithms suffer from the lack of automatic decisions for an optimal number of clusters. While trying all possible combinations is prohibitively expensive, we have got a comprehensive set of results for comparison. The performance curve on $K=1,10,20,30,40,50,60,70,100,130,150$ are shown in the Figure~\ref{fig:compair_m}.
We find $K=40$ works the best.
Table~\ref{tab:compair_m} presents results when $K=1,150,40$. \\
The comparison shows that:
\begin{enumerate}[1)]
\item Adopting a structured output space ($K=40,150$) is superior to an unstructured one($K=1$) which verifies the effectiveness of the SEC module.
\item Adopting a higher order structured output space ($K=40$) outperforms lower order one ($K=150$) which verifies the effectiveness of the HSA module.
\end{enumerate}
We also show the resulting semantic hierarchy of objects from the HSA module on VG and VRD in Figure~\ref{fig:optim_result}. Although the HSA module is not designed to sort out the objects, the unsupervised process of searching higher order connectivity patterns in the knowledge graph can contribute to an object taxonomy. At the lower levels, the object classes are classified according to more specific properties, eg. roof with railing, street with sidewalk, train with car. At the higher levels, the clusters have more abstract semantics and are classified according to more general properties, eg. glass-bottle-cup with basket-box-bag, toilet-sink-drawer with shelf-cabinet-counter.
\noindent\textbf{Clustering in HSA}.
Our behavior pattern based hierarchical clustering purely relies on the knowledge graph structure of the ground truth.
To verify the effectiveness of our clustering, we also conduct K-means clustering on word2vec embeddings of objects to obtain an external knowledge based clusters.
Table~\ref{tab:kmeans} presents the results of adopting HSA clustering results ($K=40$), adopting K-means with word2vec embedding clustering results and not adopting context clustering $K=150$).\\
The comparison shows that:
\begin{enumerate}[1)]
\item HOSE-Net with SEC($K=150$) shows comparable results to HOSE-Net with SEC and K-means with word2vec embedding($K=40$), which means, the clustering results can't improve the performance.\\
\item HOSE-Net with SEC and HSA($K=40$) outperforms HOSE-Net with SEC and K-means with word2vec embedding($K=40$), which proves that our structure-based clustering with internal relation knowledge can truly produces helpful clustering results to boost this task.
\end{enumerate}
\subsection{Comparison to State of the Art}
\textbf{Visual Genome:} Table~\ref{tab:sota_vg} shows the performance of our model outperforms the state-of-the-art methods. Our object detector is adopted from \cite{zhang2019graphical} with $mAP = 25.5$ , $IoU = 0.5$. The number of clusters for comparison is 40. These methods all adopt flat relation classifiers. VRD\cite{lu2016visual}, AsscEmbed\cite{newell2017pixels}, Frequency\cite{zellers2018neural} predict the objects and the relations without joint inference. The other works are engaged in modeling the inter-dependencies among objects and relations. MotifNet-LeftRight\cite{zellers2018neural} encodes the dependencies through bidirectional LSTMs. MSDN\cite{li2017scene},ISGG\cite{xu2017scene},KERN\cite{chen2019knowledge} rely on message passing mechanism. SGP\cite{herzig2018mapping} employs structured learning. In comparison, our framework doesn't refine the visual representations but still achieves new state-of-the-art results on SGDET, SGCLS, PRDCLS with and without graph constraint.
RelDN\cite{zhang2019graphical} proposes contrastive losses and reports Top@K Accuracy (A@K) on PredCls and SGCls in which the ground-truth subject-object pair information is also given.
We also compare with RelDN at A@K as shown in Table~\ref{tab:sota_vg}.
\noindent\textbf{VRD:} Table~\ref{tab:sota_vrd} presents results on VRD compared with state-of-the-art methods. The number of clusters for comparison is 18. The implementation details of most methods on VRD are not very clear. As shown in \cite{zhang2019graphical}, pre-training on COCO can provide stronger localization features than pre-training on ImageNet. For a fair comparison, we use the ImageNet pre-trained model. We achieve new state-of-the-art results on Relationship Detection and Phrase Detection.
\section{Conclusions}
In this work, we propose Higher Order Structure Embedded Network to address the problems caused by ignoring the structure nature of scene graphs in existing methods.
First we propose a Structure-Aware Embedding-to-Classifier module to redistribute the training samples into different classification subspaces according to the object labels and connect the subspaces with a set of context embeddings following the global knowledge graph structure.
Then we propose a Hierarchical Semantic Aggregation module to mine higher order structures of the global knowledge graph which makes the model more scalable and trainable.
\section{ACKNOWLEDGMENTS}
This work was supported by NSFC project Grant No. U1833101, SZSTI under Grant No. JCYJ20190809172201639 and the Joint Research Center of Tencent and Tsinghua.
\bibliographystyle{ACM-Reference-Format}
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