Uniformly effective Shafarevich Conjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus

Hyperelliptic Curves with Prescribed $p$-Torsion

In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the dimension of the locus of hyperelliptic curves of genus $g$ with $p$-rank at most $f$. We also produce families of curves so that the $p$-torsion of the Jacobian of each fibre contains multiple copies of the group scheme $α_p$. The method is to study curves which admit an action by $(\ZZ/2)^n$ so that the quotient is a projective line. As a result, some of these fa…

Hyperelliptic curves of genus 3 with prescribed automorphism group

We study genus 3 hyperelliptic curves which have an extra involution. The locus $Ł_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine 3-space. For every moduli point $\p \in \H_3$ such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points $\p \in \H_3$ such that $|\Aut(\p)|>4$. This is the first time that such a rational model of these curves appears in the literature.

On the $μ$-invariants of abelian varieties over function fields of positive characteristic

Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $μ$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-extension of $K$. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of $A$ and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the $μ$-invariant) in terms of other quantities including the Faltings h…

通过轨道树形表示的动态Néron-Ogg-Shafarevich准则

A Dynamical Néron--Ogg--Shafarevich Criterion via Orbital Arboreal Representations

本文通过轨道树形表示研究了非阿基米德局部域上理性映射的严格良好降解准则，揭示了残余度下降与纤维判别式在控制残余纤维的微分性中的作用，并通过具体例子验证了该准则。

通过阿尔特曼 - 豪森 - 苏斯理论对等变正规曲线进行分类

Classification of equivariantly normal curves via Altmann-Hausen-Süss theory

在特征为正的完美基域上，利用阿尔特曼 - 豪森 - 苏斯理论对具有规定商的等变正规曲线进行分类，得到同构类数量上界，并在代数闭域条件下确定其有限或无限的情况。

Pencils on real curves

We consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on Brill-Noether Theory for pencils on real curves. For coverings having topological degree 0 we introduce the covering number k and we prove the existence of coverings of degree 4 with prescribed covering number.

On metric graphs with prescribed gonality

We prove that in the moduli space of genus-g metric graphs the locus of graphs with gonality at most d has the classical dimension min{3g-3,2g+2d-5}. This follows from a careful parameter count to establish the upper bound and a construction of sufficiently many graphs with gonality at most d to establish the lower bound. Here, gonality is the minimal degree of a non-degenerate harmonic map to a tree that satisfies the Riemann-Hurwitz condition everywhere. Along the way, we establish a convenient combinatorial datum capturing such harmonic maps to trees.

Families of Log Canonically Polarized Varieties

Determining the number of singular fibers in a family of varieties over a curve is a generalization of Shafarevich's Conjecture and has implications for the types of subvarieties that can appear in the corresponding moduli stack. We consider families of log canonically polarized varieties over $¶^1$, i.e. families $g:(Y,D)\to ¶^1$ where $D$ is an effective snc divisor and the sheaf $ω_{Y/¶^1}(D)$ is $g$-ample. After first defining what it means for fibers of such a family to be singular, we show that with the addition of certain mild hypotheses (the fibers have finite automorphism group, $\sO_…

线性系统中光滑轨迹的 Brauer 群与平面曲线雅可比上的 torsors

Brauer groups of smooth loci in linear systems and torsors over Jacobians of plane curves

研究单连通光滑射影簇线性系统中光滑轨迹的 Brauer 群，在适当丰沛条件下证明其至多为 Z/2Z，并应用于计算通用光滑平面曲线相对雅可比上的 Tate-Shafarevich 群。

无额外自同构的指定p-秩的Artin-Schreier曲线

Artin-Schreier curves of prescribed p-rank without extra automorphisms

该研究针对奇素数p，构造了无额外自同构的指定p-秩的Artin-Schreier曲线，确定了自同构群仅为Cp的精确数值范围，通过Valentini-Madan定理及相关界完成例外族排除。

K3 surfaces with interesting groups of automorphisms

By the fundamental result of I.I. Piatetsky-Shapiro and I.R. Shafarevich (1971), the automorphism group Aut(X) of a K3 surface X over C and its action on the Picard lattice S_X are prescribed by the Picard lattice S_X. We use this result and our method (1980) to show finiteness of the set of Picard lattices S_X of rank $\ge 3$ such that the automorphism group Aut(X) of the K3 surface X has a non-trivial invariant sublattice S_0 in S_X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic lattices S_0 it has been proved by the author before (1980, 1995). Thus we extend thi…