For $r\geq 3$ 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 $[\overline{ \mathcal{ R}}_g^r]$ in $\mathrm{ Pic}_\mathbb{ Q}(\overline{ \mathcal{ R}}_g)$. Furthermore, we examine a strongly Brill-Noether divisor in $\overline{ \mathcal{ M}}_{ g-1,2}$: we show its irreducibility and compute some of its coefficients in $\mathrm{ Pic}_\mathbb{ Q}(\overline{ \mathcal{ M}}_{ g-1,2})$. As a consequence of our results, the moduli space $\mathcal{ R}_{ 14,2}$ is of general type.
Categories: math.AG