The Integral Chow Ring of the Stack of Pointed Hyperelliptic Curves

TL;DR

This paper computes the integral Chow ring of the stack of pointed hyperelliptic curves, providing complete results for small numbers of points and partial results for larger n, with implications for moduli spaces of genus 2 curves.

Contribution

It offers the first explicit calculations of the integral Chow ring for the stack of pointed hyperelliptic curves, advancing understanding of their intersection theory.

Findings

Complete Chow ring for n=1,2

Partial Chow ring results for 3≤n≤2g+2

Implications for the moduli space of genus 2 curves

Abstract

We study the integral Chow ring of the stack $\mathcal{H}_{g,n}$ parametrizing $n$-pointed smooth hyperelliptic curves of genus $g$. We compute the integral Chow ring of $\mathcal{H}_{g,n}$ for $n=1,2$ completely, while for $3\leq n\leq2g+2$ we compute it up to the additive order of a single class in degree 2. We obtain partial results also for $n=2g+3$. In particular, taking $g=2$ and recalling that $\mathcal{H}_{2,n}=\mathcal{M}_{2,n}$, our results hold for $\mathrm{CH}^*(\mathcal{M}_{2,n})$ for $1\leq n\leq7$.