The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

TL;DR

This paper computes the rational Chow ring of hyperelliptic moduli spaces with marked points, showing it is generated by tautological classes and establishing their rationality for certain parameters.

Contribution

It determines the Chow ring structure for $ leq 2g+6$ points and proves the rationality of $ leq 3g+5$ point hyperelliptic moduli spaces.

Findings

Chow ring of $ leq 2g+6$ points explicitly computed.

Chow ring of partial compactification generated by tautological divisors.

Hyperelliptic moduli spaces are rational for $ leq 3g+5$ points.

Abstract

We determine the rational Chow ring of the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$ when $n \leq 2g+6$. We also show that the Chow ring of the partial compactification $\mathcal{I}_{g,n}$, parametrizing $n$-pointed irreducible nodal hyperelliptic curves, is generated by tautological divisors. Along the way, we improve Casnati's result that $\mathcal{H}_{g,n}$ is rational for $n \leq 2g+8$ to show $\mathcal{H}_{g,n}$ is rational for $n \leq 3g+5$.