Sylvain Maugeais
Let $g$ and $g'$ be two integers and $p$ a prime number. Denote by $\mathscr H_{g, g', p}^c$ the moduli space of morphisms of degree $p$ between smooth curves of genus $g$ and $g'$ and with constant ramification. The purpose of this article is to construct and describe a modular compactification of this space.
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M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann-Hurwitz formula, and consider an example suggested by S. Saito.
We construct several modular compactifications of the Hurwitz space $H^d_{g/h}$ of genus $g$ curves expressed as $d$-sheeted, simply branched covers of genus $h$ curves. These compactifications are obtained by allowing the branch points of the covers to collide to a variable extent. They are very well-behaved if $d = 2, 3$, or if relatively few collisions are allowed. We recover as special cases the spaces of twisted admissible covers of Abramovich, Corti and Vistoli and the spaces of hyperelliptic curves of Fedorchuk.
Hurwitz correspondences are certain multivalued self-maps of the moduli space $\mathcal{M}_{0,N}$. They arise in the study of Thurston's topological characterization of rational functions. We consider the dynamics of Hurwitz correspondences and ask: On which compactifications of $\mathcal{M}_{0,N}$ should they be studied? We compare a Hurwitz correspondence $\mathcal{H}$ across various modular compactifications of $\mathcal{M}_{0,N}$, and find a weighted stable curves compactification $X_N^\dagger$ that is optimal for its dynamics. We use $X_N^\dagger$ to show that the $k$th dynamical degree o…
We revisit Ekedahl, Lando, Shapiro and Vainshtein's compactification of the stack of simply ramified covers of the projective line except for a fixed ramification profile above infinity. In particular we draw a connection with the Harris and Mumford stack of admissible covers showing that the boundary of the ELSV compactification appears as a contraction of the boundary of the stack of admissible covers. This lays the needed foundations for a combinatoral interpretation of boundary points of the ELSV compactification
Logarithmic Hurwitz Spaces in Mixed and Positive Characteristic with Wild Ramification
本文提出混合与正特征中带有对数结构的Hurwitz空间，用于研究特征0和等特征下曲线分枝覆盖的紧化问题，并证明其log平滑性。
We describe a sequence of effective divisors on the Hurwitz space $H_{d,g}$ for $d$ dividing $g-1$ and compute their cycle classes on a partial compactification. These divisors arise from vector bundles of syzygies canonically associated to a branched cover. We find that the cycle classes are all proportional to each other.
The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.
We investigate two families of divisors which we expect to play a distinguished role in the global geometry of Hurwitz space. In particular, we show that they are extremal and rigid in the small degree regime $d \leq 5$. We further show their significance in the problem of computing the sweeping slope of Hurwitz space in these degrees. In the process, we prove various general results about the divisor theory of Hurwitz space, including a proof of the independence of the boundary components of the admissible covers compactification. Some basic open questions and further directions are discussed…
Hurwitz spaces and Inverse Galois Theory
本文综述了Hurwitz空间在反 Galois 理论中的五十年研究，重点介绍近期进展，包括大域、紧化、降理论和模塔等半现代发展，以及通过研究组件环重塑Hurwitz空间算术的最新成果。
The classical Hurwitz spaces, that parameterize compact Riemann surfaces equipped with covering maps to ${\mathbb P}_1$ of fixed numerical type with simple branch points, are extensively studied in the literature. We apply deformation theory, and present a study of the Kähler structure of the Hurwitz spaces, which reflects the variation of the complex structure of the Riemann surface as well as the variation of the meromorphic map. We introduce a generalized Weil-Petersson Kähler form on the Hurwitz space. This form turns out to be the curvature of a Quillen metric on a determinant line bundle…
In this paper we describe explicit generating functions for a large class of Hurwitz-Hodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves, and the orbifold Gromov-Witten theory of DM stacks. Our main tool is Atiyah-Bott localization: its underlying philosophy is to transla…
We use tropical and nonarchimedean geometry to study the moduli space of genus $0$ stable maps to $\mathbb{P}^1$ relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this toric variety may be interpreted as a moduli space for tropical relative stable maps with the same discrete data. As a consequence, we confirm an expectation of Bertram and the first two authors, that the tropical Hurwitz cycles are tropicalizations of classical Hurwitz cycles. As a second application, we obtain a full descendant correspondence for genus $0$ r…
The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko-Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP^1 by CP^1 with prescribed ramification points and ramification types is related to the degree of the Lyashko--Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko-Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations …
We investigate the universal Jacobian of degree n line bundles over the Hurwitz stack of double covers of P^1 by a curve of genus g. Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification of this universal Jacobian; a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of these stacks in the cases when n-g is even. An important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms. This parametrization generaliz…
Hurwitz spaces, Nichols algebras, and Igusa zeta functions
本文构造次数 4、5 的 Hurwitz 空间拟映射紧化，建立算术统计、Nichols 代数与 Igusa zeta 函数的精确联系，并回答相关计数与拓扑起源问题。
A finite étale map between irreducible, normal varieties is called tame, if it is tamely ramified with respect to all partial compactifications whose boundary is the support of a strict normal crossings divisor. We prove that if the Galois group of a Galois covering contains a normal nilpotent subgroup of index bounded by a constant N, then the covering is tame if and only if it is tamely ramified with respect to a single distinguished partial compactification only depending on N. The main tools used in the proof are Temkin's local purely inseparable uniformization and a Lefschetz type theorem…
Boundary stratifications of Hurwitz spaces
本研究建立了Hurwitz空间紧化的不可约边界层与装饰树的组合对应关系，由此定义了加细的tropical Hurwitz空间并给出计算实现，为复动力系统等领域提供了新的组合与几何工具。
Chow Rings of Hurwitz Spaces with Marked Ramification
本文研究带标记分歧点的 Hurwitz 空间紧化的 Chow 环，通过将边界层表示为粘合映射的像并证明低次数情形 Chow 环平凡，证明了当 k=3 时余维 2 Chow 群由边界层基类生成。
Canonical heights, periods and the Hurwitz zeta function
本文为相对丰富的对数典范线丛引入典范高度，证明其与 K-稳定性及 Kähler-Einstein 度量高度的关系，并用周期极限公式显式计算算术对数曲面等情形，应用于志村曲线整模型和 Arakelov 度量界。
For a closed oriented surface $ Σ$ let $X_{Σ,n}$ be the space of isomorphism classes of orientation preserving $n$-fold branched coverings $ Σ\rightarrow S^2 $ of the two-dimensional sphere. At a previous paper, the authors constructed a compactification $\bar{X}_{Σ,n}$ of the space that coincides with the Diaz-Edidin-Natanzon-Turaev compactification of the Hurwitz space $H(Σ,n)\subset X_{Σ,n}$ consisting of isomorphism classes of branched coverings with all critical values being simple. Using Grothendieck's dessins d'enfants we construct a cell structure of the compactification. The obtained …
We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs $(S, D)$ where $S$ is a degeneration of $\mathbb{P}^1 \times \mathbb{P}^1$ and $D \subset S$ is a degeneration of a curve of class $(3,3)$. We prove that the compactified moduli space is a smooth Deligne--Mumford stack with 4 boundary components. We relate it to the moduli space of genus 4 curves; we show that it compactifies the blow-up of the hyperelliptic locus. We also relate it to a compactification of the Hurwitz space of triple coverings of $\math…
Tropicalizing polynomial strata
研究次数$D\geq2$复多项式分歧层${\rm Poly}_D(\vec\mu^*)$的热带化，通过构建带框装饰多项式树空间$\mathcal{T}_D(\vec\mu^*)$，利用多种方法解决热带化难题，证明其与相关骨架同构，实现紧致化并与其他紧致化比较。
Hodge classes on the moduli space of W(E_6)-covers and the geometry of A_6
本文确定 W(E_6)-覆盖 Hurwitz 空间上对应 Weyl 群25个不可约表示的 Hodge 类，并给出 A_6 经 W(E_6) 型 Prym-Tyurin 簇单值化的初等证明。
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