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 $3\le n\le 2g+2$ 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 ${\mathrm{C}\mathrm{H}}^{\ast }({\mathcal{M}}_{2,n})$ for $1\le n\le 7$.