$A_r$-stable curves and the Chow ring of $\overline{\mathcal{M}}_3$

TL;DR

This paper introduces a new moduli stack for $A_r$-stable curves and uses it to compute the Chow ring of the moduli space of genus 3 curves, providing explicit algebraic invariants.

Contribution

It defines the moduli stack $ ilde{rak{M}}_{g,n}^r$ of $A_r$-stable curves and computes the Chow rings of $ar{rak{M}}_3$ and $ ilde{rak{M}}_3^7$, advancing understanding of their algebraic structure.

Findings

Computed the Chow ring of $ar{rak{M}}_3$.

Computed the Chow ring of $ ilde{rak{M}}_3^7$.

Introduced the moduli stack $ ilde{rak{M}}_{g,n}^r$.

Abstract

In this work, we introduce the moduli stack $\widetilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed, $A_r$-stable curves of genus $g$ and use it to compute the Chow ring of $\overline{\mathcal{M}}_3$. As a byproduct, we also compute the Chow ring of $\widetilde{\mathcal{M}}_3^7$. All the Chow rings are assumed to be with coefficients in $\mathbb{Z}[1/6]$.