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

We study the moduli space ${\mathcal{A}\mathcal{S}}_g$ 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 ${\mathcal{A}\mathcal{S}}_g$ is connected for all possible $g$. When $p>3$, it turns out that ${\mathcal{A}\mathcal{S}}_g$ 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 ${\mathcal{A}\mathcal{S}}_g$.