Modular compactifications of ℳ2,n with Gorenstein curves

Battistella, Luca

doi:10.2140/ant.2022.16.1547

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Modular compactifications of $\mathcal{M}_{2,n}$ with Gorenstein curves

Luca Battistella

Vol. 16 (2022), No. 7, 1547–1587

DOI: 10.2140/ant.2022.16.1547

Abstract

We study the geometry of Gorenstein curve singularities of genus two, and their stable limits. These singularities come in two families, corresponding to either Weierstrass or conjugate points on a semistable tail. For every $1 \leq m < n$ , a stability condition — using one of the markings as a reference point, and thus not $𝔖_{n}$ -symmetric — defines proper Deligne–Mumford stacks $\bar{ℳ} {2, n}^{(m)}$ with a dense open substack representing smooth curves.

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Keywords

moduli of curves, Gorenstein singularities, curves of genus two, crimping spaces

Mathematical Subject Classification

Primary: 14H10

Secondary: 14H20, 14H45

Milestones

Received: 15 November 2019

Revised: 5 October 2021

Accepted: 2 November 2021

Published: 16 October 2022

Authors

Luca Battistella
Mathematisches Institut Ruprecht-Karls-Universität Heidelberg Heidelberg Germany

Vol. 16, No. 7, 2022