The class of the Prym-Brill-Noether divisor

Abstract

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 $[{\overline{\mathcal{R}}}_g^r]$ 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.

arXiv comment25 pages. To appear in Journal of the Institute of Mathematics of Jussieu

References in corpus (4)

• Skeletons of Prym varieties and Brill--Noether theory
• ${\overline{\mathcal{R}}}_{15}$ is of general type
• Brill-Noether theory for cyclic covers
• Prym-Brill-Noether Loci of special curves