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

Abstract

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 $Pic^{g-1}C$ which are linearly equivalent to $2\Theta$. The embedded tangent space at a semi-stable non-stable bundle $\xi\oplus\xi^{-1}$, where $\xi$ is a degree zero line bundle, is shown to consist of those divisors in $|2\Theta|$ which contain $Sing(\Theta_{\xi})$ where $\Theta_{\xi}$ is the translate of $\Theta$ by $\xi$. We also obtain geometrical results on the structure of this tangent space.