Connectedness of The Moduli Space of Artin-Schreier Curves of Fixed Genus

TL;DR

This paper investigates the structure and connectedness of the moduli space of Artin-Schreier curves of fixed genus over fields of positive characteristic, revealing conditions under which the space is connected and relating combinatorial data to its geometry.

Contribution

It constructs deformations of Artin-Schreier curves to analyze the moduli space's stratification and proves connectedness results for different characteristics and genera.

Findings

The moduli space is connected for all genera when p=3.

For p>3, the space is connected for sufficiently large genus.

Answers a question relating combinatorial graphs to the geometry of the moduli space.

Abstract

We study the moduli space $\mathcal{AS}_{g}$ of Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of positive characteristic $p$. The moduli space is partitioned by irreducible strata, where each stratum parameterizes Artin-Schreier curves whose ramification divisors have the same coefficients. We construct deformations of these curves to study the relations between those strata. As an application, when $p=3$, we prove that $\mathcal{AS}_{g}$ is connected for all possible $g$. When $p>3$, it turns out that $\mathcal{AS}_{g}$ is connected for sufficiently large value of $g$. In the course of our work, we answer Pries and Zhu's question about how a combinatorial graph determines the geometry of $\mathcal{AS}_g$.