The Integral Chow Ring of the Stack of Pointed Hyperelliptic Curves

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 $1\leq n\leq7$.