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

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.

The paper is one of the four papers that compose the author's PhD thesis. In particular, it contains Section 1.1, Section 1.3 and Section 1.4 of arXiv:2211.09793. Some typos have been corrected and the exposition was improved. 35 pages; comments are very welcome