Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

On theta divisors of vector bundles

I will explain recent work with V. B. Mehta on theta divisors of vector bundles. On a compact Riemann surface of genus g >= 2 (or an non-singular projective curve) there is a notion of a theta divisor: the locus of line bundles M of degree zero such that a fixed line bundle L of degree g-1 admits non-zero maps from M --> L. This locus is classically known as the theta divisor. In higher rank, i.e, when we replace L by a vector bundle V of slope g-1 the situation is quite different. In particular there are vector bundles of slope g-1 (even stable) which do not admit such theta divisors. In any case a general semi-stable bundle does admit a theta divisor. But we do not have any description of this locus. In the present work we provide an open set of semi-stable vector bundles with theta divisors.