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Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space ofsemi-stable vector bundles of rank 3 and trivial determinant over $C$, and by$J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$has a canonical theta divisor $\Theta$. The space $SU_C(3)$ is a double coverof $P^8=|3\Theta|$ branched along a sextic hypersurface, the Coble sextic. Inthe dual $\check{P}^8=|3\Theta|^*$, where $J^1$ is embedded, there is a uniquecubic hypersurface singular along $J^1$, the Coble cubic. We prove that thesetwo hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, bylooking at some special linear sections of these hypersurfaces, we can observeand reinterpret some classical results of algebraic geometry in a context ofvector bundles: the duality of the Segre-Igusa quartic with the Segre cubic,the symmetric configuration of 15 lines and 15 points, the Weddle quarticsurface and the Kummer surface.Comment: 21 pages. Supersedes math.AG/0408318. To appear in Internat. J. Mat

VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO