Singularities of 2Θ-divisors in the Jacobian

We Consider the linear system \2theta(0)\ of second order theta functions over the Jacobian JC of a non-hyperelliptic curve C. A result by J. Fay says that a divisor D is an element of \2theta(0)\ contains the origin O is an element of JC with multiplicity 4 if and only if D contains the surface C - C = {O(p - q) \ p, q is an element of C} subset of JC. In this paper we generalize Fay's result and some previous work by R.C. Gunning. More precisely, we describe the relationship between divisors containing O with multiplicity 6, divisors containing the fourfold C-2 - C-2 = {O(p + q - r - s) \ p, q, r, s is an element of C}, and divisors singular along C - C, using the third exterior product of the canonical space and the space of quadrics containing the canonical curve. Moreover we show that some of these spaces are equal to the linear span of Brill-Noether loci in the moduli space of semi-stable rank 2 vector bundles with canonical determinant over C, which can be embedded in \2theta(0)\.