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Logarithmic Hurwitz Spaces in Mixed and Positive Characteristic with Wild Ramification

Matthias Hippold

TL;DR

The paper develops a framework for logarithmic Hurwitz spaces in mixed and positive characteristic with wild ramification by constructing liftable moduli spaces $\mathcal{LH}_{A}^{\mathbb{Z}_{(p)}}$ and $\mathcal{LH}_{A,\Xi}^{\mathbb{F}_p}$ and describing their modular data via Artin–Schreier covers and differential forms. It introduces auxiliary moduli spaces for exact and quasi-exact differential forms, studies the Cartier operator, and uses tropical/logarithmic data to organize deformations and liftings. Key results include that all irreducible components of the special fiber have dimension $3g-3+N$, and that the initial cases $p=2$, $g=0$ yield log smooth moduli over the base ($\mathbb{Z}_{(2)}$ or $\mathbb{F}_2$). The paper also provides explicit constructions and dimension counts for the Artin–Schreier and differential-form moduli, and analyzes a degree-2 example on $\mathbb{P}^1$ ramified at four points to illustrate the geometry and the necessity of exactness/quasi-exactness conditions. This framework offers a path to systematic study and potential generalization to higher genus and degree, enriching the theory of Hurwitz spaces in mixed/positive characteristic.

Abstract

We introduce new logarithmic Hurwitz spaces $\mathcal{LH}^{\mathbb{Z}_{(p)}}_A$ and $\mathcal{LH}^{\mathbb{F}_{p}}_{A,Ξ}$ over $\mathbb{Z}_{(p)}$ and $\mathbb{F}_p$ respectively that in the mixed characteristic case can be considered as a compactification of the admissible cover stack parametrizing ramified covers of curves in characteristic $0$ of degree $p$ and in the equicharacteristic case compactify the space of separable maps between smooth curves of degree $p$. These Hurwitz spaces will carry a logarithmic structure and to emphasize that they are informative, we prove that in some first cases our Hurwitz spaces are log smooth. To achieve this, we introduce various Moduli spaces that parametrize Artin-Schreier covers and the locus of zeroes and poles of certain differential forms, show their smoothness and compute their dimension.

Logarithmic Hurwitz Spaces in Mixed and Positive Characteristic with Wild Ramification

TL;DR

The paper develops a framework for logarithmic Hurwitz spaces in mixed and positive characteristic with wild ramification by constructing liftable moduli spaces and and describing their modular data via Artin–Schreier covers and differential forms. It introduces auxiliary moduli spaces for exact and quasi-exact differential forms, studies the Cartier operator, and uses tropical/logarithmic data to organize deformations and liftings. Key results include that all irreducible components of the special fiber have dimension , and that the initial cases , yield log smooth moduli over the base ( or ). The paper also provides explicit constructions and dimension counts for the Artin–Schreier and differential-form moduli, and analyzes a degree-2 example on ramified at four points to illustrate the geometry and the necessity of exactness/quasi-exactness conditions. This framework offers a path to systematic study and potential generalization to higher genus and degree, enriching the theory of Hurwitz spaces in mixed/positive characteristic.

Abstract

We introduce new logarithmic Hurwitz spaces and over and respectively that in the mixed characteristic case can be considered as a compactification of the admissible cover stack parametrizing ramified covers of curves in characteristic of degree and in the equicharacteristic case compactify the space of separable maps between smooth curves of degree . These Hurwitz spaces will carry a logarithmic structure and to emphasize that they are informative, we prove that in some first cases our Hurwitz spaces are log smooth. To achieve this, we introduce various Moduli spaces that parametrize Artin-Schreier covers and the locus of zeroes and poles of certain differential forms, show their smoothness and compute their dimension.
Paper Structure (15 sections, 30 theorems, 97 equations, 3 figures)

This paper contains 15 sections, 30 theorems, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Preliminaries
  4. Log Curves
  5. Hurwitz Spaces in Characteristic 0
  6. Exact and Quasi-Exact Differential Forms
  7. The Moduli Spaces LHZA and LHFAXi
  8. Definition and Representability
  9. First Properties of LHURphgnZ
  10. Moduli Spaces of Relevant Objects
  11. The Moduli Space AScovhen of Artin--Schreier Covers with Fixed Ramification Divisor
  12. The Moduli Space of Exact Differential Forms M0nexm
  13. The Moduli Space of Quasi-Exact Differential Forms M0nquexm
  14. Logarithmic Smoothness of LHZA and LHFAXi for p=2 and g=0
  15. Example: Degree Two Covers of P1 Ramified over Four Points

Key Result

Theorem 2.6

The Moduli space $\mathcal{M}_{g,n}^{log}$ of log curves with $n$ markings and genus $g$ is represented by the Moduli stack of nodal curve $\overline{\mathcal{M}}_{g,n}$ with logarithmic structure associated to the boundary divisor coming from the locus of nodal curves.

Figures (3)

  • Figure 1: The generic dual level graph $\Delta_{1;0}$ of the source curve of one irreducible component of the special fiber of our Moduli space. Note that when the four markings are labeled $0,1,\lambda$ and $\infty$, the elliptic curve is glued to the point $\sqrt{\lambda}$.
  • Figure 2: The dual level graph $\Delta_{1;0,0}$ of the target curve of three strata of the special fiber of our Moduli space.
  • Figure 3: The two graphs $\Delta_{1;0;0,0}$ and $\Delta_{0,0;0;0}$ corresponding to strata in the boundary of the strata described by $\Delta_{1;0,0}$.

Theorems & Definitions (84)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 74 more