The local geometry of the stack of $A_r$-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 $r=5$.

41 pages, 7 figures, comments are welcome