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

Abstract

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.

25 pages, to appear in Coll. Mathematica. The authors are indebted to the referee for the careful reading and valuable comments which have certainly improved the readability of the paper. Authors have been supported by Mission Sustainability 2017 - FamCurves (CUP E81-18000100005 Tor Vergata University) and PRIN 2015EYPTSB-011-Geomety of Algebraic varieties (Node Tor Vergata)