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

TL;DR

This paper investigates the local structure and deformation theory of the stack of pointed $A_r$-stable curves, classifying points and substacks, and exploring conditions for good moduli spaces.

Contribution

It provides a detailed classification of points and substacks of the $A_r$-stable curves stack, and studies obstructions to the existence of good moduli spaces.

Findings

Classified all closed points of the stack of $A_r$-stable curves.

Identified open substacks of the moduli stack of degree 2 cyclic covers with separated good moduli spaces.

Constructed an example of a non-projective good moduli space for $A_5$-stable curves.

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 $r=5$.