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

Luca Battistella

Vol. 16 (2022), No. 7, 1547–1587
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 m < n, a stability condition — using one of the markings as a reference point, and thus not 𝔖n-symmetric — defines proper Deligne–Mumford stacks ¯2,n(m) with a dense open substack representing smooth curves.

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