Complete classification of irreducible components of the Brill-Noether locus of rank-2 vector bundles of degree d and speciality 2 on a general nu-gonal curve

In this paper we consider Brill-Noether loci of rank-two, stable vector bundles of given degree d, with speciality 2, on a general ν-gonal curve C of genus g with the aim of studying their irreducible components in the whole range of interest for d, namely 2g−2≤d≤4g−4. We prove that such a Brill-Noether locus either is empty or we completely classify all of its irreducible components, giving also some extra information about their dimensions (exhibiting both regular and superabundant components), about their birational geometry as well as giving precise description of their general point. As a by-product of our general results, we deduce also some consequences on Brill-Noether loci in rank two with fixed general determinant line bundle instead of fixed degree.