Brill-Noether loci of rank-two vector bundles on a general n-gonal curve

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general ν-gonal curve. We classify its irreducible components having at least expected dimension. We moreover describe the general member F of such components just in terms of extensions of line bundles with suitable "minimality properties", providing information on the birational geometry of such components as well as on the very-ampleness of F