Brill-Noether loci of rank two vector bundles on a general $\nu$-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 $\nu$-gonal curve. We classify its reduced components whose dimensions are at least the corresponding Brill-Noether number. We moreover describe the general member $\mathcal{F}$ of such components just in terms of extensions of line bundles with suitable {\em minimality properties}, providing information on the birational geometry of such components as well as on the very-ampleness of $\mathcal{F}$.