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 $[\widetilde C / C; x_1, \dots, x_m]$ where $[C, x_1, \dots, x_m]$ is an $m$--pointed curve of genus $g$ and $\widetilde C/ C$ is an \'etale 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 $\mathrm{CH}^\bullet(\mathcal{R}_{g})$ 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 $[\widetilde C/ C]$ such that, if $C \rightarrow E$ is the bi--elliptic structure, the composition $\widetilde C \rightarrow E$ factors through an elliptic cover of $E$. Our main contribution is thus the non--tautology of the class $[\mathcal{R}\mathcal{B}_8^0] \in \mathrm{CH}^*(\mathcal{R}_8)$. In the course of establishing this theorem, a similar result for the compact moduli spaces $\overline{\mathcal{R}}_{g; 2m}$ for $g + m \geq 8$ is proven.