The tangent space to the moduli space of vector bundles on a curve and thesingular locus of the theta divisor of the jacobian

We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on Pic( 9-1)C that are linearly equivalent to 2-. The embedded tangent space at a s emistable non-stable bundle xi + xi (-1), where xi is a degree zero line bu ndle, is shown to consist of those divisors in \2-1\ that contain Sing (-xi ) where -xi is the translate of - by xi. We also obtain geometrical results on the structure of this tangent space.