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

Deopurkar, Anand; Han, Changho

doi:10.1090/tran/8225

Stable log surfaces, admissible covers, and canonical curves of genus 4
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by Anand Deopurkar and Changho Han PDF

Trans. Amer. Math. Soc. 374 (2021), 589-641 Request permission

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.

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