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 ) (S, D) where S S is a degeneration of P 1 × P 1 \mathbb {P}^1 \times \mathbb {P}^1 and D ⊂ S D \subset S is a degeneration of a curve of class ( 3 , 3 ) (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 P 1 \mathbb {P}^1 by genus 4 curves.