Logarithmic Hurwitz Spaces in Mixed and Positive Characteristic with Wild Ramification

TL;DR

This paper develops new logarithmic Hurwitz spaces over mixed and positive characteristic fields, providing compactifications and demonstrating their log smoothness, with applications to ramified covers and Artin-Schreier covers.

Contribution

Introduction of new logarithmic Hurwitz spaces in mixed and positive characteristic with proofs of their log smoothness and related moduli spaces.

Findings

Logarithmic Hurwitz spaces are constructed over $Z_{(p)}$ and $F_p$.

Proved log smoothness of these Hurwitz spaces in some cases.

Computed dimensions of moduli spaces parametrizing Artin-Schreier covers.

Abstract

We introduce new logarithmic Hurwitz spaces $\mathcal{LH}^{\mathbb{Z}_{(p)}}_A$ and $\mathcal{LH}^{\mathbb{F}_{p}}_{A,\Xi}$ 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.