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arXiv · 2603.29853

The local geometry of the stack of $A_r$-stable curves

Davide Gori·Ludvig Modin·Michele Pernice
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Abstract

In this paper we study the local geometry of the stack of pointed $A_r$-stable curves. In particular, we analyze the deformation theory of $A_r$-stable curves and their automorphism groups in order to study the combinatorics of families of curves over $[\mathbb{A}^1/\mathbb{G}_m]$, and use this to classify all closed points of the stack of $A_r$-stable curves. As a byproduct, we also classify all open substacks of the moduli stack of degree $2$ cyclic covers of $\mathbb{P}^1$ that admit a separated good moduli space. This is the first in a series of three papers aimed at studying obstructions for the existence of good moduli spaces for stacks of curves with $A$-type singularities, and using these to find an open substack of the stack of $A_r$-stable curves that admits a proper non-projective good moduli space when $r=5$.

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Davide Gori, Ludvig Modin, Michele Pernice. 2026-07-13. The local geometry of the stack of $A_r$-stable curves. https://arxiv.org/abs/2603.29853

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