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, ${\mathcal{R}\mathcal{H}}_g$ and ${\widetilde{\mathcal{R}\mathcal{H}}}_g$, 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 ${\mathcal{R}\mathcal{H}}_g^n$ and ${\widetilde{\mathcal{R}\mathcal{H}}}_g^n$ for $n=1,\dots ,\lfloor (g+1)/2\rfloor$. In this paper we present quotient stack descriptions of the components ${\mathcal{R}\mathcal{H}}_g^n$ for even $g$ and compute their integral Chow rings, thereby completing the computation for all irreducible components of ${\mathcal{R}\mathcal{H}}_g$. In addition, we give quotient stack presentations for all irreducible components of ${\widetilde{\mathcal{R}\mathcal{H}}}_g$ and determine when the rigidification map ${\widetilde{\mathcal{R}\mathcal{H}}}_g^n\to {\mathcal{R}\mathcal{H}}_g^n$ is a root gerbe. We then use this to compute the Chow rings of ${\widetilde{\mathcal{R}\mathcal{H}}}_g^n$ 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.

24 pages, comments welcome!