Stable log surfaces, trigonal covers, and canonical curves of genus 4

Abstract

We describe a compactification of the moduli space of pairs $(S, C)$ where $S$ is isomorphic to $\PP^1 \times \PP^1$ and $C \subset S$ is a genus 4 curve of class $(3,3)$. We show that the compactified moduli space is a smooth Deligne-Mumford stack with 4 boundary components. We relate our compactification with compactifications of the moduli space $\mathcal M_4$ of genus 4 curves. In particular, we show that our space compactifies the blow-up of the hyperelliptic locus in ${\mathcal M}_4$. We also relate our compactification to a compactification of the Hurwitz space ${\mathcal H}^3_4$ of triple coverings of $\PP^1$ by genus 4 curves.