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

TL;DR

This paper computes the integral Chow rings of specific components of the moduli stacks of hyperelliptic Prym pairs for odd genus, advancing understanding of their algebraic structure.

Contribution

It provides explicit presentations and Chow rings for the components of hyperelliptic Prym stacks and hyperelliptic Spin curves of odd genus, extending prior work.

Findings

Chow rings of hyperelliptic Prym components computed

Presentations for moduli stacks of hyperelliptic Spin curves obtained

Chow ring of unordered pairs of divisors in P^1 calculated

Abstract

This paper is the second in a series devoted to describing the integral Chow ring of the moduli stacks $\mathcal{RH}_g$ of hyperelliptic Prym pairs. For fixed genus $g$, the stack $\mathcal{RH}_g$ is the disjoint union of $\lfloor (g+1)/2 \rfloor$ components $\mathcal{RH}_g^n$ for $n = 1, \ldots, \lfloor (g+1)/2 \rfloor$. In this paper, we give presentations and compute the integral Chow rings of the components $\mathcal{RH}_g^{(g+1)/2}$ for odd $g$. As an application, we also obtain presentations and Chow rings for all irreducible components of the moduli stack of hyperelliptic Spin curves of odd genus. An intermediate result of independent interest is the computation of the integral Chow ring of the moduli stack of unordered pairs of divisors of the same even degree in $\mathbb{P}^1$.