The class of the Prym-Brill-Noether divisor

For $r\ge 3$ and $g=\frac{r(r+1)}{2}$, we study the Prym-Brill-Noether variety $V^r(C,\eta )$ associated to Prym curves $[C,\eta ]$. The locus ${\mathcal{R}}_g^r$ in ${\mathcal{R}}_g$ parametrizing Prym curves $(C,\eta )$ with nonempty $V^r(C,\eta )$ is a divisor. We compute some key coefficients of the class $[{\bar{\mathcal{R}}}_g^r]$ in ${\mathrm{P}\mathrm{i}\mathrm{c}}_{\mathbb{Q}}({\bar{\mathcal{R}}}_g)$. Furthermore, we examine a strongly Brill-Noether divisor in ${\bar{\mathcal{M}}}_{g-1,2}$: we show its irreducibility and compute some of its coefficients in ${\mathrm{P}\mathrm{i}\mathrm{c}}_{\mathbb{Q}}({\bar{\mathcal{M}}}_{g-1,2})$. As a consequence of our results, the moduli space ${\mathcal{R}}_{14,2}$ is of general type.