Hyperelliptic 𝐴ᵣ-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 M ¯ 3 {M}₃, the moduli stack of stable curves of genus 3 3. In this paper, we introduce the moduli stack H ~ g r {H}gʳ of hyperelliptic A r Aᵣ -stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic A r Aᵣ -stable curves. In particular, we prove that H ~ g r {H}gʳ 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 Mgʳ (the moduli stack of A r Aᵣ -stable curves) as the closure of the moduli stack of smooth hyperelliptic curves.