Moduli spaces of bundles and of Hilbert schemes over $\nu$-gonal curve

The aim of this paper is two--fold. We first strongly improve our previous main result published in Proc. Math. Am. Soc., concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees $d$, with at least $d-2g+4$ independent global sections, on a general $\nu$--gonal curve $C$ of genus $g$. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which turn out to be special and stable.