Linear series on moduli spaces of vector bundles on curves.

The present work develops a geometric study of linear series and generalized theta functions on moduli spaces of vector bundles on curves, with the aim of understanding effective numerical statements in the spirit of higher dimensional geometry. We give effective base point freeness and projective normality bounds for pluritheta linear series, as well as dimension bounds for the base loci of determinant linear series. This study has an abelian and a nonabelian part. For the nonabelian part the main technique is focused on giving upper bounds on the dimension of Quot schemes, via constructions known as elementary transformations. On the other hand, from the abelian point of view, we introduce the notion of Verlinde bundle on the Jacobian of a curve, and study this type of bundle with methods specific to the theory of vector bundles on abelian varieties. As a byproduct of these techniques we obtain a new global picture on duality for generalized theta functions and we formulate some further conjectures on optimal effective statements.