Cubic symmetroids and vector bundles on a quadric surface

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.Comment: 6 pages; Comments welcom