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

TL;DR

This paper describes a detailed compactification of a moduli space of log surfaces and relates it to the moduli space of genus 4 curves, revealing new geometric structures and boundary components.

Contribution

It explicitly constructs a smooth compactification of a moduli space of log surfaces of Picard rank 2 and connects it to genus 4 curve moduli spaces.

Findings

The compactified moduli space is a smooth Deligne--Mumford stack with 4 boundary components.

It provides a compactification of the blow-up of the hyperelliptic locus in genus 4 curves.

It relates the moduli space to a compactification of the Hurwitz space of triple coverings of P^1 by genus 4 curves.

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.