Stefan Wewers
This note gives a survey on some results related to the stable reduction of three point covers, which were the topic of my talk at the {\em Journées Arithmetiques} 2003 in Graz.
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R. Coleman and K. McMurdy compute the stable reduction of $X_0(p^3).$ On the basis of their ideas, we compute the stable reduction of $X_0(p^4).$ As a result, in the stable reduction of $X_0(p^4)$, we find irreducible components, defined by $a^p-a=t^{p+1}$. These components are called Deligne-Lusztig curve for ${\rm SL}_2(\mathbb{F}_p).$ We also compute the intersection multiplicity datum in the stable reduction of $X_0(p^4)$.
We compute the stable reduction of the Lubin-Tate space $X(π^2)$ in the equal characteristic case, on the basis of Coleman-McMurdy's ideas. Namely, in this paper, we actually construct a stable covering of $X(π^2).$ This paper also includes intersection multiplicity datum of the stable model of $X(π^2)$. Our method is very explicit and elementary.
We study Galois covers of the projective line branched at three points with bad reduction to characteristic p, under the condition that p exactly divides the order of the Galois group. As an application of our results, we prove that the field of moduli of such a cover is at most tamely ramified at p.
We determine the stable reduction at $p$ of all three point covers of the projective line with Galois group ${\rm SL}_2(p)$. As a special case, we recover the results of Deligne and Rapoport on the reduction of the modular curves $X_0(p)$ and $X_1(p)$. Our method does not use the fact that modular curves are moduli spaces. Instead, we rely on results of Raynaud and the authors which describe the stable reduction of three point covers whose Galois group is strictly divisible by $p$.
Raynaud gave a criterion for a branched $G$-cover of curves defined over a mixed-characteristic discretely valued field $K$ with residue characteristic $p$ to have good reduction in the case of either a three-point cover of $\mathbb{P}^1$ or a one-point cover of an elliptic curve. Specifically, such a cover has potentially good reduction whenever $G$ has a Sylow $p$-subgroup of order $p$ and the absolute ramification index of $K$ is less than the number of conjugacy classes of order $p$ in $G$. In the case of an elliptic curve, we generalize this to the case in which $G$ has an arbitrarily lar…
Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion subgroup on the Jacobian of $C$. We also determine divisibility of line bundles on $C$, including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of $C$.
Semistable reduction of smooth quartics
研究离散赋值域上光滑平面四次曲线的半稳定约化，利用几何不变量理论和Bruhat-Tits建筑上的稳定性函数方法构造接近稳定的特殊纤维平面模型，并证明非超椭圆情形下GIT稳定平面模型与稳定模型的关系。
Semistable Reduction of Plane Quartics
研究平面四次曲线稳定模型计算问题，建立抽象稳定模型与GIT - 稳定平面模型联系，证明GIT - 稳定平面模型存在条件，明确稳定模型与主导态射性质，为计算稳定模型提供几何框架。
We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties, and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general set-up for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley-Weil theorem for stacks.
In this paper we study the semi-stable reduction of $p$ and $p^2$-cyclic covers of curves in equal characteristic $p>0$. The main tool we use is the classical Artin-Schreier-Witt theory for $p^n$-cyclic covers in characteristic $p$. Although the results of this paper concern only the cases of degree $p$ and $p^2$-cyclic cover we develop the techniques and the framework in which the general cyclic case can be studied.
The local obstruction to semi-stable reduction for abelian varieties
本文综述了格罗滕迪克定义的阿贝尔簇半稳定约化局部障碍群的研究进展，并首次纳入等特征局部域情形，旨在给出仅依赖维数的群论刻画。
Michel Raynaud gave a criterion for a three-point G-cover f : Y \rightarrow X = P^1, defined over a p-adic field K, to have good reduction. In particular, if the order of a p-Sylow subgroup of G is p, and the number of conjugacy classes of elements of order p is greater than the absolute ramification index e of K, then f has potentially good reduction. We give a different proof of this criterion, which extends to the case where G has an arbitrarily large cyclic p-Sylow subgroup, answering a question of Raynaud. We then use the criterion to give a family of examples of three-point covers with g…
In his Ph. D. thesis, C. Lehr offers an algorithm which gives the stable model for p-cyclic covers of the projective line over a p-adic field under the conditions that the branch locus whose cardinal is m+1 has the so called equidistant geometry and m<p. In this note we give an algorithm also in the equidistant geometry case but without condition on m. In particular we are able to study the reduction at 2 of hyperelliptic curves with equidistant branch locus.
The Classification of the Stable Marked Reduction of Genus 2 Curves in Residue Characteristic 2
研究二阶超椭圆曲线在余特征2下的稳定缩减结构，通过计算机代数系统对特定曲线进行分类。
We prove that the Vologodsky integral of a mermorphic one-form on a curve over a $p$-adic field with semi-stable reduction restrict to Coleman integrals on the rigid subdomains reducing to the components of the smooth part of the special fiber and that on the connecting annuli the differences of these Coleman integrals form a harmonic cochain on the edges of the dual graph of the special fiber. This determines the Vologodsky integral completely. We analyze the behavior of the integral on the connecting annuli and we explain the results in the case of a Tate elliptic curve.
Let K be the function field of a connected regular scheme S of dimension 1, and let f : X -> Y be a finite cover of projective smooth and geometrically connected curves over K with g(X) greater or equal to 2. Suppose that f can be extended to a finite cover {\mathcal X} -> {\mathcal Y} of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). We prove that there then exists a (unique) minimal such cover. This gives a canonical way to extend X -> Y to a finite cover of semi-stable models over S.
Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p>0$, and let $K^+$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^+$-schemes $X=\text{Spec}(R)$ with semi-stable reduction to the Galois cohomology of the fundamental group of the geometric generic fibre $π_1(X_{\bar{K}})$ with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings' theory of almost étale extensions.
Computing the Stable Reduction of Hyperelliptic Curves in Residue Characteristic 2
本文研究了特征零域上超椭圆曲线在剩余特征 2 下的稳定约化问题，证明了标定曲线稳定约化的一般性质，改进了现有计算算法，并给出了直至亏格 g=30 的具体算例。
For a dynamical system on n-dimensional projective space over a number field or a function field, we show that semi-stable reduction implies the minimality of the resultant. We use this to show that every such dynamical system over a number field admits a globally minimal presentation.
We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.
We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a hypergeometric differential equation. This generalizes the result of Deligne- Rapoport on the reduction of the modular curve X(p).
Semistable reduction of covers of degree $p$
本文提出一种计算局部域上配备 $p$ 次态射到射影直线的曲线半稳定约化的方法，涵盖非伽罗瓦覆盖，并特别实现了 $p=3$ 平面四次曲线在 SageMath 中的约化，基于 Berkovich 解析几何与微分函数。
The Deligne-Mumford stable reduction theorem asserts that for a family of stable curves over the punctured disk, after a finite base change, the family can be completed in a unique way to a family of stable curves over the disk. In this survey we discuss stable reduction theorems in a number of different contexts. This includes a review of recent results on abelian varieties, canonically polarized varieties, and singularities. We also consider the semi-stable reduction theorem and results concerning simultaneous stable reduction.
In this paper we study the reduction of Galois covers of curves, from characteristic 0 to characteristic p. The starting point is a is a recent result of Raynaud which gives a criterion for good reduction for covers of the projective line branch at 3 points. Under some condition on the Galois group, we extend this criterion to the case of 4 branch points. Moreover, we describe the reduction of the Hurwitz space of such covers and compute the number of covers with good reduction.
Line bundles on perfectoid covers: case of good reduction
本文研究完美oid空间的Picard群与Picard函子，证明良约化极限覆盖的Picard函子可由特殊纤维表示，并借此解决若干公开问题，同时构造了$\mathbb G_m$的Hodge--Tate谱序列。
Special covers are metacyclic covers of the projective line, with Galois group of order pm, which have a specific type of bad reduction to characteristic p. Such covers arise in the study of the arithmetic of Galois covers of the projective line with three branch points. Our results provide a classification of all special covers in terms of certain lifting data in characteristic p.
In this paper, we define syntomic complex for modulus pair (X,D), where X is regular semi-stable family and D is an effective Cartier divisor on X. We compute its cohomology sheaves.
In this paper we study the reduction of $p$-cyclic covers of the $p$-adic line ramified at exactly four points. For $p=2$ these covers are elliptic curves and Deuring has given a criterion for when such a curve has good reduction. Here we consider the case of $p>2$ and completely determine the stable model of the cover. In particular we obtain a finite extension $R'$ of $R$ necessary for the stable reduction to be defined. No additional conditions are imposed on the geometry of the branch locus and thus this work can be viewed as a first step towards understanding the situation where branch po…
We answer negatively an open problem of Illusie on the DR-decomposability of the log de Rham complex of the special fiber of a semi-stable reduction over the Witt ring. We also show that $E_1$ degeneration of the Hodge to log de Rham spectral sequence does not imply DR-decomposability of semi-stable varieties.
The semi-simplicity of the Hodge group is proved for a simple Abelian variety with a stable reduction of odd toric (reductive) rank. If, besides, the dimension of the Abelian variety is an odd integer, then the Hodge conjecture on algebraic cycles holds for it.
We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to T. Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain purel…
In this paper we study the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal characteristics (0,p). We are also able to describe the Galois action on these covers in terms of some geometric and combinatorial datas in characteristic p endowed with the action of the Galois group of the residue field.
A cheap method for constructing canonical models and complete moduli for complex projective varieties with a structure called "rational plurifibration" is given. A result about semistable reduction (whose nature is slightly different from alg-geom/9707012) is deduced. The method involves recursive application of moduli stacks of twisted stable maps (math.AG/9908167) and a systematic circumvention of any difficult MMP issues.
Stable Reduction via the Log Canonical Model
该研究提出了推广德利涅-芒福德稳定约化到高维的猜想，在极小模型纲领两个标准猜想下证明其在大特征下成立，并恢复了哈肯-科瓦奇定理关于特定稳定曲面模叠层真理性的结论。
The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider compactifications of this space. By allowing to parameterize as well curves with controlled singularities (the so called stable curves), Deligne and Mumford constructed a projective compactification. The properness of this compactification translates into the stable reduction theorem that they prove, its projectivity is a later theorem of Knudsen and Mumford. Th…
The stable reduction theorem says that a family of curves of genus $g\geq 2$ over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new proof of this result for curves defined over $\mathbb{C}$ using the Kähler-Einstein metrics on the fibers to obtain the limiting stable curves at the punctures.
We give a new proof of the Semistable Reduction Theorem for curves. The main idea is to present a curve $Y$ over a local field $K$ as a finite cover of the projective line $X=\PP^1_K$. By successive blowups (and after replacing $K$ by a suitable finite extension) we construct a semistable model of $X$ whose normalization with respect to the cover is a semistable model of $Y$.
Let C/K: F = 0 be a smooth plane quartic over a complete discrete valuation field K. In a previous paper the authors togetehr with Q. Liu give various characterizations of the reduction (i.e. non-hyperelliptic genus 3 curve, hyperelliptic genus 3 curve or bad) of the stable model of C: in terms of the existence of a special plane quartic model and in terms of the valuations of the Dixmier-Ohno invariants of C. The last one gives in particular an easy computable criterion for the reduction type. However, it does not produce a stable model, even in the case of good reduction. In this paper we gi…
We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. We study minimality and semi-stability, considering what conditions imply minimality (Theorem 4.4) and whether semi-stable models and presentations are minimal, proving results in the degree two case (Theorems 4.6, 4.7). We prove the singular reduction of a semi-stable pre…
In this paper we compute the multiple cover Gromov-Witten integrals (analog of the Aspinwall-Morrison formula) for the unramified compactification of the moduli space of stable maps to an embedded $\PO$ in a Calabi-Yau threefold $X$ with the normal bundle $\mathcal{O}_{\PO}(-1) \bigoplus \mathcal{O}_{\PO}(-1)$.
Let $K$ be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second $\ell$-adic cohomology of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga-Satake Abelian varieties. On our way, we settle existence and termination of certain semi-stable flops in mixed characteristic, and study gr…
On the Kodaira types of elliptic curves with potentially good supersingular reduction
本文研究了在扩张 $L/K$ 为度为 $2$ 的野分歧且曲线 $E/K$ 具有潜在良好超奇异约化时，对椭圆曲线 Kodaira 类型施加的限制，并分析了同源椭圆曲线的可能约化类型。
We study the deformation of $G$-marked stable curves in the case where $G$ is a cyclic group, and construct a parameterizing space for $G$-marked stable curves of a given numerical type. This is then used in order to study the components of the locus of stable curves admitting the action of a cyclic group of non prime order $d$, extending work of F. Catanese in the case where $d$ is prime.
The small $p$-adic Simpson correspondence in the semi-stable reduction case
本文将光滑形式概型上的小 Simpson 对应已知结果推广至半稳定情形，为可提升半稳定形式概型建立了 Hitchin-small integral $v$-bundles 与 Higgs bundles 范畴及对应模栈的等价关系。
In this paper, we introduce the notion of parabolic stable pairs on Calabi-Yau 3-folds and invariants counting them. By applying the wall-crossing formula developed by Joyce-Song, Kontsevich-Soibelman, we see that they are related to generalized Donaldson-Thomas invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds. Consequently, the conjectural multiple cover formula of generalized DT invariants is shown to be equivalent to a certain product expansion formula of the generating series of parabolic stable pair invariants. The application of this result to the multiple cov…
Suppose F=W(k)[1/p] where W(k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p>2. We construct integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0,p). This modification of Breuil's theory results in the following application in the spirit of Shafarevich's Conjecture. If Y is a projective algebraic variety over the field of rational numbers with good reduction modulo all primes different from 3 and semi-stable reduction modulo 3 then for the Hodge numbers of the complexification…
The main aim of this paper is to show that a cyclic cover of $\mathbb{P}^n$ branched along a very general divisor of degree $d$ is not stably rational provided that $n \ge 3$ and $d \ge n+1$. This generalizes the result of Colliot-Thélène and Pirutka. Generalizations for cyclic covers over complete intersections and applications to suitable Fano manifolds are also discussed.
Let K be a complete field of unequal characteristics $(0,p)$. The aim of this paper is to describe the the semi-stable models for covers $\bold P^1_K@>>>\bold P^1_K$ of degree p, unramified outside $r\leq p$ points and totally ramified above one of them, under the assumption that the ramified locus has a particular reduction type (which always occurs if $r\leq4$). We are principally concerned with the minimal semi-stable models which separate the ramified fibers.
Deformations of Zappatic stable surfaces and their Galois covers
本文研究可形变为具有唯一En型奇点的平面Zappatic稳定曲面的代数曲面，证明其Galois覆盖在n≥4时为一般型单连通曲面并给出Chern数公式，进而证明n>30时此类曲面不存在。
We study the conductor of Picard curves over $\mathbb{Q}$, which is a product of local factors. Our results are based on previous results on stable reduction of superelliptic curves that allow to compute the conductor exponent $f_p$ at the primes $p$ of bad reduction. A careful analysis of the possibilities of the stable reduction at $p$ yields restrictions on the conductor exponent $f_p$. We prove that Picard curves over $\mathbb{Q}$ always have bad reduction at $p=3$, with $f_3\geq 4$. As an application we discuss the question of finding Picard curves with small conductor.
Admissible covers and stable maps
本文研究目标曲线的容许覆盖与稳定映射模空间在 $\overline{M}_{g,n}$ 上产生的闭链，证明统一公式以恢复并细化 ELSV 公式及 Gromov-Witten/Hurwitz 对应，并完成低亏格计算验证。
This is a resubmission of preprint 9401008 , which has some TeXnical errors introduced by the "reform" procedure (designed to avoid precisely these problems!). The original can be formatted by editing out the messages "%% following line cannot be broken..." at line 1674. This problem has been corrected in the current version. The content of the paper has not been changed in any way. My aplogies for any inconvenience caused. SB
Adic curves: stable reduction, skeletons and metric structure
本文研究任意秩仿射域上 adic 曲线的结构，通过引入 ranger 紧化这一新工具，分类曲线上的点、证明半稳定约化定理、定义曲线骨架，并构造其上的度量结构与形变收缩。
Let $E_G$ be a stable principal $G$--bundle over a compact connected Kaehler manifold, where $G$ is a connected reductive linear algebraic group defined over the complex numbers. Let $H\subset G$ be a complex reductive subgroup which is not necessarily connected, and let $E_H\subset E_G$ be a holomorphic reduction of structure group. We prove that $E_H$ is preserved by the Einstein-Hermitian connection on $E_G$. Using this we show that if $E_H$ is a minimal reductive reduction in the sense that there is no complex reductive proper subgroup of $H$ to which $E_H$ admits a holomorphic reduction o…
We use twisted sheaves to study the problem of index reduction for Brauer classes. In general terms, this problem may be phrased as follows: given a field $k$, a $k$-variety $X$, and a class $α\in \Br(k)$, compute the index of the class $α_{k(X)} \in \Br(X)$ obtained from $α$ by extension of scalars to $k(X)$. We give a general method for computing index reduction which refines classical results of Schofield and van den Bergh. When $X$ is a curve of genus 1, we use Atiyah's theorem on the structure of stable vector bundles with integral slope to show that our formula simplifies dramatically, g…
A key tool in the study of algebraic surfaces and their moduli is Brieskorn's simultaneous resolution for families of algebraic surfaces with simple (du Val or ADE) singularities. In this paper we show that a similar statement holds for families of curves with at worst simple (ADE) singularities. For a family $\mathscr X\to B$ of ADE curves, we give an explicit and natural resolution of the rational map $B\to \bar M_g$. Moreover, we discuss a lifting of this map to the moduli stack $ \bar {\mathcal M}_g$, i.e. a simultaneous semi-stable reduction for the family $\mathscr X/B$. In particular, w…
We define a proper moduli stack for degree $p$ covers $f:Y \to \cX$ where $\cX$ is a twisted stable curve in the sense of [5] and [4], and $Y$ is a stable curve which via $f$ is a torsor over $\cX$ under a finite flat group scheme $\cG \to \cX$.
A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus $g$ curves with $n$ labeled points $\modmgn$. Here the generic fibre of $p$ has genus $g$ and intersects $C$ in $n$ points. In this paper we give an efficient method to calculate this homology class. We apply this to any polynomial in two complex variables $p :\bc^2\to\bc$ where the $n$ points on a fibre are its points at infinity.
In this paper, we study tame Galois coverings of semistable models that arise from torsion points on elliptic curves. These coverings induce Galois morphisms of intersection graphs and we express the decomposition groups of the edges in terms of the reduction type of the elliptic curve. To that end, we first define the reduction type of an elliptic curve $E/K(C)$ on a subgraph of the intersection graph $Σ(\mathcal{C})$ of a strongly semistable model $\mathcal{C}$. In particular, we define the notions of good and multiplicative reduction on subgraphs of the intersection graph of $Σ(\mathcal{C})…
Stable rational cohomology splitting of moduli of branched covers of curves
本文研究了曲线分支覆盖模空间的稳定有理上同调，给出了其关于中间覆盖次数的分解，并用因子化同调及带标签的构型空间加以解释，证明了 d=4,5 时稳定奇维上同调类的存在。
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