Verlinde Bundles and Generalized Theta Linear Series

The main purpose of this paper is to provide a new approach to the study of generalized theta linear series on moduli spaces of vector bundles on a smooth complex projective curve X, based on vector bundle techniques on abelian varieties. To this effect we naturally associate to these linear series what will be called the Verlinde vector bundles on the Jacobian of X and make use of some special properties that they enjoy. At least technically the present work should be interpreted as following two directions interacting with each other. The first two sections are dedicated to an extensive study of the Verlinde bundles with the goal of explaining why they emerge as a useful tool in various problems related to pluritheta line bundles and duality for generalized theta functions. The remaining sections contain an analysis of the geometric properties of pluritheta line bundles, with the motivation of finding effective bounds for global generation and normal generation via the above mentioned vector bundle techniques.