Moduli of stable sheaves on a smooth quadric and a Brill–Noether locus

We prove that the moduli space of stable sheaves of rank 2 with the Chern classes c1=OQ(1,1) and c2=2 on a smooth quadric Q in P3 is isomorphic to P3. Using this identification, we give a new proof that a Brill–Noether locus, defined as the closure of the stable bundles with at least three linearly independent sections, on a non-hyperelliptic curve of genus 4, is isomorphic to the Donagi–Izadi cubic threefold.