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 \(\mathbb{P}^1 \times \mathbb{P}^1\) and \(D \subset S\) is a degeneration of a curve of class \((3,3)\). We prove...

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Bibliographic Details
Published in:arXiv.org 2020-06
Main Authors: Deopurkar, Anand, Han, Changho
Format: Article
Language:English
Subjects:
Degeneration
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1807.08413