Cubic symmetroids and vector bundles on a quadric surface

TL;DR

This paper studies special rank 2 vector bundles on a quadric surface, describing their jumping conics as cubic symmetroids, and shows these conics uniquely determine the bundles, with the moduli space being rational.

Contribution

It characterizes the set of jumping conics for these vector bundles as cubic symmetroids and proves their uniqueness and the rationality of the moduli space.

Findings

Jumping conics form a cubic symmetroid in projective space.

The set of jumping conics uniquely determines the vector bundle.

The moduli space of these bundles is rational.

Abstract

We investigate the jumping conics of stable vector bundles $\Ee$ of rank 2 on a smooth quadric surface $Q$ with the Chern classes $c_1=\Oo_Q(-1,-1)$ and $c_2=4$ with respect to the ample line bundle $\Oo_Q(1,1)$. We describe the set of jumping conics of $\Ee$, a cubic symmetroid in $\PP_3$, in terms of the cohomological properties of $\Ee$. As a consequence, we prove that the set of jumping conics, $S(\Ee)$, uniquely determines $\Ee$. Moreover we prove that the moduli space of such vector bundles is rational.