Local structure of SU_C(3) for a curve of genus 2

The aim of this note is to give a precise description of the local structure of the moduli space of rank 3 vector bundles over a curve of genus 2, which is in particular shown to be a local complete intersection. This allows us to investigate the local structure of the branch locus of the theta map, whose dual is known to be the Coble cubic in P^8.