The moduli stack of $A_r$-stable curves

TL;DR

This paper introduces a new moduli stack of $A_r$-stable curves, extending classical results and describing the structure of curves with specific singularities, as part of a series studying the Chow ring of genus 3 curves.

Contribution

It defines the moduli stack $ ilde{M}_{g,n}^r$ of $A_r$-stable curves and extends classical results, including the existence of contraction morphisms and the description of certain substacks.

Findings

Defined the moduli stack of $A_r$-stable curves.

Extended classical results to the new stack.

Described the normalization of substacks with $A_h$-singularities.

Abstract

This paper is the first in a series of four papers aiming to describe the (almost integral) Chow ring of $\bar{\mathcal{M}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we introduce the moduli stack $\tilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed $A_r$-stable curves and extend some classical results about $\bar{\mathcal{M}}_{g,n}$ to $\tilde{\mathcal{M}}_{g,n}^r$, namely the existence of the contraction morphism. Moreover, we describe the normalization of the locally closed substack of $\tilde{\mathcal{M}}_{g,n}^r$ parametrizing curves with $A_h$-singularities for a fixed $h\leq r$.