Self-duality of Coble’s quartic hypersurface and applications

The moduli space M0 of semi-stable rank 2 vector bundles with fixed trivial determinant over a non-hyperelliptic curve C of genus 3 is isomorphic to a quartic hypersurface in P7 (Coble’s quartic). We show that M0 is self-dual and that its polar map associates to a stable bundle E ∈ M0 a bundle F which is characterized by dimH 0(C,E ⊗F ) = 4. The projective space PH0(C,E ⊗ F ) is equipped with a net of quadrics Π and it is shown that the map which associates to E ∈ M0 the isomorphism class of the plane quartic Hessian curve of Π is a dominant map to the moduli space of genus 3 curves.