Subvarieties of the hyperelliptic moduli determined by group actions

Abstract

Let ${\mathcal{H}}_g$ be the moduli space of genus $g$ hyperelliptic curves. In this note, we study the locus $\mathcal{L}$ in ${\mathcal{H}}_g$ of curves admitting a $G$-action of given ramification type $\sigma$ and inclusions between such loci. For each genus we determine the list of all possible groups, the inclusions among the loci, and the corresponding equations of the generic curve in $\mathcal{L}$. The proof of the results is based solely on representations of finite subgroups of $PG{L}_2(\mathbb{C})$ and the Riemann-Hurwitz formula.