Jumping conics on a smooth quadric in $\PP_3$

TL;DR

This paper studies the geometric properties of stable rank 2 vector bundles on a smooth quadric surface, focusing on the hypersurfaces formed by their jumping conics and describing related moduli spaces.

Contribution

It characterizes the set of jumping conics as a hypersurface of degree c_2(E)-1 and describes moduli spaces for lower second Chern class cases.

Findings

Jumping conics form a hypersurface of degree c_2(E)-1 in projective space.

The structure of moduli spaces is described for lower c_2(E).

Provides geometric insights into stable vector bundles on quadrics.

Abstract

We investigate the jumping conics of stable vector bundles $E$ of rank 2 on a smooth quadric surface $Q$ with the first Chern class $c_1=\Oo_Q(-1,-1)$ with respect to the ample line bundle $\Oo_Q(1,1)$. We show that the set of jumping conics of $E$ is a hypersurface of degree $c_2(E)-1$ in $\PP_3^*$. Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on $Q$ in the cases of lower $c_2(E)$.