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

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 ℙ1 × ℙ1 and D ⊂ 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 ℙ1 by genus 4 curves.