  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.
