A Hurwitz divisor on the Moduli of Prym curves

Abstract

For genus $g=2i\ge 4$ and the length $g-1$ partition $\mu =(4,2,\dots ,2,-2,\dots ,-2)$ of 0, we compute the first coefficients of the class of $\overline{D}(\mu )$ in ${\mathrm{Pic}}_{\mathbb{Q}}({\overline{\mathcal{R}}}_g)$, where $D(\mu )$ is the divisor consisting of pairs $[C,\eta ]\in {\mathcal{R}}_g$ with $\eta \cong {\mathcal{O}}_C(2x_1+x_2+\cdots +x_{i-1}-x_i-\cdots -x_{2i-1})$ for some points $x_1,\dots ,x_{2i-1}$ on $C$. We further provide several enumerative results that will be used for this computation.

arXiv commentFinal version. Accepted in Geometriae Dedicata