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

TL;DR

This paper investigates the tangent space structure of the moduli space of rank two vector bundles on a curve, revealing its relation to the singular locus of the theta divisor on the Jacobian, and completes a proof of a key embedding property.

Contribution

It completes the proof that the moduli space embeds into a linear system on the Picard variety and characterizes the tangent space at semi-stable bundles in terms of theta divisor singularities.

Findings

Tangent space at semi-stable bundles consists of divisors containing the singular locus of translated theta divisors.

The embedding of the moduli space into the linear system of divisors on Pic^{g-1}C is fully established.

Geometrical insights into the structure of the tangent space are provided.

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.