Jumping conics on a smooth quadric in ${\text{PP}}_3$

We investigate the jumping conics of stable vector bundles $E$ of rank 2 on a smooth quadric surface $Q$ with the first Chern class $c_1={\text{Oo}}_Q(-1,-1)$ with respect to the ample line bundle ${\text{Oo}}_Q(1,1)$. We show that the set of jumping conics of $E$ is a hypersurface of degree $c_2(E)-1$ in ${\text{PP}}_3^{\ast }$. Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on $Q$ in the cases of lower $c_2(E)$.