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

Abstract

We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs $(S,D)$ where $S$ is a degeneration of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and $D\subset S$ is a degeneration of a curve of class $(3,3)$. We prove that the compactified moduli space is a smooth Deligne--Mumford stack with 4 boundary components. We relate it to the moduli space of genus 4 curves; we show that it compactifies the blow-up of the hyperelliptic locus. We also relate it to a compactification of the Hurwitz space of triple coverings of ${\mathbb{P}}^1$ by genus 4 curves.

49 pages, 9 figures. Added more details and improved the introduction. To appear in Transactions of the American Mathematical Society