Non--tautological cycles on Prym moduli spaces

We denote by ${\mathcal{R}}_{g;m}$ the moduli space of $m$–pointed Prym curves of genus $g$, that is, tuples $[\tilde{C}\mathrm{/}C;x_1,\dots ,x_m]$ where $[C,x_1,\dots ,x_m]$ is an $m$–pointed curve of genus $g$ and $\tilde{C}\mathrm{/}C$ is an étale double cover of $C$. In this paper, we address the problem of the non–tautology of the Chow ring of ${\mathcal{R}}_{g;m}$. The locus which allows us to achieve earlier bounds for the non–tautology of ${\mathrm{C}\mathrm{H}}^{\bullet }({\mathcal{R}}_g)$ compared to ${\mathcal{M}}_g$ is the component $\mathcal{R}{\mathcal{B}}_g^0$ of the locus of bi–elliptic Prym curves. This parametrises covers $[\tilde{C}\mathrm{/}C]$ such that, if $C\to E$ is the bi–elliptic structure, the composition $\tilde{C}\to E$ factors through an elliptic cover of $E$. Our main contribution is thus the non–tautology of the class $[\mathcal{R}{\mathcal{B}}_8^0]\in {\mathrm{C}\mathrm{H}}^{\ast }({\mathcal{R}}_8)$. In the course of establishing this theorem, a similar result for the compact moduli spaces ${\bar{\mathcal{R}}}_{g;2m}$ for $g+m\ge 8$ is proven.