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

Abstract

Let C be a curve of genus two. We denote by [Formula: see text] the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by Jd the variety of line bundles of degree d on C. In particular, J1 has a canonical theta divisor Θ. The space [Formula: see text] is a double cover of ℙ8 = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual [Formula: see text], where J1 is embedded, there is a unique cubic hypersurface singular along J1, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.