Heisenberg invariant quartics and SU_C(2) for a curve of genus four

Abstract

If C is a curve of genus 4 without vanishing theta-nulls then there exists a unique (irreducible) Heisenberg-invariant quartic Q_C in |2\Theta| = P^{15} such that Sing Q_C contains the image of SU_C(2), the moduli space of rank 2 vector bundles with trivial determinant. Moreover, in each eigen-P^7 of the Heisenberg action on |2\Theta|, Q_C restricts to the classical Coble quartic of the corresponding Prym-Kummer variety. We compare Q_C with the hypersurface G_3 in |2\Theta| of divisors containing a translate of C in J(C), and show that in the eigen-P^7s G_3 recovers Beauville--Debarre's quadrisecant planes of the Prym-Kummers (this works for any genus). Using the Recillas construction this enables us to deduce, contrary to the analogous result for genus 3, that Q_C and G_3 are distinct.