Twists of Mukai bundles and the geometry of the level $3$ modular variety over $\overline{\mathcal{M}}_8$

Abstract

For a curve $C$ of genus $6$ or $8$ and a torsion bundle $\eta$ of order $\ell$ we study the vanishing of the space of global sections of the twist $E_C \otimes \eta$ of the rank two Mukai bundle $E_C$ of $C$. The bundle $E_C$ was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Pl\"ucker embedding. Globalizing the vanishing condition, we obtain divisors on the moduli spaces $\overline{\mathcal{R}}_{6,\ell}$ and $\overline{\mathcal{R}}_{8,\ell}$ of pairs $[C, \eta]$. First we characterize these divisors by different conditions on linear series on the level curves, afterwards we calculate the divisor classes. As an application, we are able to prove that $\overline{\mathcal{R}}_{8,3}$ is of general type.