Sur l'espace des modules des fibres vectoriels de rang 3 sur une courbe de genre 2 et la cubique de Coble

Abstract

In this thesis we have proved a conjecture about the moduli space SU_X(3) of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve X, due to I. Dolgachev. Given X a smooth projective curve of genus 2, and the embedding of the jacobian JX into |3\Theta|, A. Coble proved, at the begining of XX century, that there exists a unique cubic hypersurface C in |3\Theta|* \simeq P^8, JX[3]-invariant and singular along JX. On the other hand, we have a map of degree 2 from SU_X(3) over |3\Theta|, ramified along a sextic hypersurface B. The Dolgachev's conjecture affirms that the sextic B is the dual variety of the Coble's cubic C.