  The aim of this paper is two--fold. We first strongly improve our previous
main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of
rank two vector bundles on a general $\nu$-gonal curve"), 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.
