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Stable log surfaces, admissible covers, and canonical curves of genus 4
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2020-10-20 , DOI: 10.1090/tran/8225
Anand Deopurkar , Changho Han

We describe a compactification of the moduli space of pairs $(S, C)$ where $S$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$ and $C \subset S$ is a genus 4 curve of class $(3,3)$. We show that the compactified moduli space is a smooth Deligne-Mumford stack with 4 boundary components. We relate our compactification with compactifications of the moduli space $\mathcal M_4$ of genus 4 curves. In particular, we show that our space compactifies the blow-up of the hyperelliptic locus in ${\mathcal M}_4$. We also relate our compactification to a compactification of the Hurwitz space ${\mathcal H}^3_4$ of triple coverings of $\mathbb{P}^1$ by genus 4 curves.

中文翻译:

4 属的稳定对数曲面、可容许覆盖和规范曲线

我们描述了对 $(S, C)$ 模空间的紧化,其中 $S$ 同构于 $\mathbb{P}^1 \times \mathbb{P}^1$ 并且 $C \subset S$ 是类 $(3,3)$ 的属 4 曲线。我们表明紧缩模空间是一个平滑的 Deligne-Mumford 堆栈,具有 4 个边界分量。我们将我们的紧化与属 4 曲线的模空间 $\mathcal M_4$ 的紧化联系起来。特别是,我们表明我们的空间压缩了 ${\mathcal M}_4$ 中超椭圆轨迹的膨胀。我们还将我们的紧化与通过属 4 曲线对 $\mathbb{P}^1$ 的三重覆盖的 Hurwitz 空间 ${\mathcal H}^3_4$ 的紧化联系起来。
更新日期:2020-10-20
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