IP^r scrolls arising from Brill-Noether theory and K3-surfaces

In this paper we study examples of P(r)-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill-Noether theory of the general curve in the primitive linear system on S and from some results of Lazarsfeld. We show that such scrolls form an open dense subset of a component H of their Hilbert scheme; moreover, we study some properties of H (e.g. smoothness, dimensional computation, etc.) just in terms of F(g), the moduli space of such K3' s, and M(v)(S), the moduli space of semistable torsion-free sheaves of a given rank on S. One of the motivation of this analysis is to try to introducing the use of projective geometry and degeneration techniques in order to studying possible limits of semistable vector-bundles of any rank on a very general K3 as well as Brill-Noether theory of vector-bundles on suitable degenerations of projective curves. We conclude the paper by discussing some applications to the Hilbert schemes of geometrically ruled surfaces introduced and studied in Calabri et al. (Rend LinceiMat Appl 17(2): 95-123, 2006) and Calabri et al. (Rend Circ Mat Palermo 57(1): 1-32, 2008).