The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs III

Abstract: This paper is the third and final part of a series devoted to the description of the integral Chow rings of the moduli stacks of hyperelliptic Prym pairs. For a fixed genus $g$, there are two natural stacks, $\mathcal{RH}_g$ and $\widetilde{\mathcal{RH}}_g$, parametrizing hyperelliptic Prym pairs, with the former being the $\mu_2$-rigidification of the latter. Both decompose as the disjoint union of $\lfloor (g+1)/2 \rfloor$ components, denoted $\mathcal{RH}_g^n$ and $\widetilde{\mathcal{RH}}_g^n$ for $n = 1, \ldots, \lfloor (g+1)/2 \rfloor$. In this paper we present quotient stack descriptions of the components $\mathcal{RH}_g^n$ for even $g$ and compute their integral Chow rings, thereby completing the computation for all irreducible components of $\mathcal{RH}_g$. In addition, we give quotient stack presentations for all irreducible components of $\widetilde{\mathcal{RH}}_g$ and determine when the rigidification map $\widetilde{\mathcal{RH}}_g^n \to \mathcal{RH}_g^n$ is a root gerbe. We then use this to compute the Chow rings of $\widetilde{\mathcal{RH}}_g^n$ for all $g$ and $n$, with the sole exception of the case where $g$ is odd and $n=(g+1)/2$. Finally, in the appendix, we discuss $G$-gerbes induced by an homomorphism of abelian groups $H \to G$ and an $H$-gerbe.