Hyperelliptic 𝐴ᵣ-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 M ¯ 3 \overline {\mathcal {M}}_3 , the moduli stack of stable curves of genus 3 3 . In this paper, we introduce the moduli stack H ~ g r \widetilde {\mathcal {H}}_g^r of hyperelliptic A r A_r -stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic A r A_r -stable curves. In particular, we prove that H ~ g r \widetilde {\mathcal {H}}_g^r is a smooth algebraic stack that can be described using cyclic covers of twisted curves of genus 0 0 and it embeds in M ~ g r \widetilde {\mathcal M}_g^r (the moduli stack of A r A_r -stable curves) as the closure of the moduli stack of smooth hyperelliptic curves.