Hyperelliptic $A_r$-stable curves (and their moduli stack)

TL;DR

This paper introduces the moduli stack of hyperelliptic $A_r$-stable curves, generalizing hyperelliptic stable curves, and proves its smoothness and embedding properties within the broader moduli of $A_r$-stable curves.

Contribution

It defines and studies the properties of the moduli stack of hyperelliptic $A_r$-stable curves, extending the theory to include cyclic covers of twisted genus 0 curves.

Findings

$ ilde{oldsymbol{H}}_g^r$ is a smooth algebraic stack.

$ ilde{oldsymbol{H}}_g^r$ embeds as the closure of hyperelliptic curves in $ ilde{oldsymbol{M}}_g^r$.

The stack can be described via cyclic covers of twisted genus 0 curves.

Abstract

This paper is the second in a series of four papers aiming to describe the (almost integral) Chow ring of $\Mbar_3$, the moduli stack of stable curves of genus $3$. In this paper, we introduce the moduli stack $\Htilde_g^r$ of hyperelliptic $A_r$-stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic $A_r$-stable curves. In particular, we prove that $\Htilde_g^r$ is a smooth algebraic stacks which can be described using cyclic covers of twisted curves of genus $0$ and it embeds in $\Mtilde_g^r$ (the moduli stack of $A_r$-stable curves) as the closure of the moduli stack of smooth hyperelliptic curves.