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

This paper is the second in a series of four papers aiming to describe the (almost integral) Chow ring of ${\text{Mbar}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we introduce the moduli stack ${\text{Htilde}}_g^r$ 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 ${\text{Htilde}}_g^r$ is a smooth algebraic stacks which can be described using cyclic covers of twisted curves of genus $0$ and it embeds in ${\text{Mtilde}}_g^r$ (the moduli stack of $A_r$-stable curves) as the closure of the moduli stack of smooth hyperelliptic curves.