Effective base point freeness on moduli spaces of vector bundles on curves

We will present a few results on effective bounds for the global generation of multiples of generalized theta line bundles on moduli spaces of vector bundles on a curve X. The techniques involved are of two different flavors: on one hand we will use a result (of independent interest) giving an optimal upper bound on the dimension of the Hilbert scheme of coherent quotients of a fixed vector bundle, while on the other hand we will appeal to general vector bundle techniques on abelian varieties via the notion of Verlinde bundle on the Jacobian of X. If time allows we will briefly address problems related to normal generation and base points for these linear series.