Special linear systems and syzygies

Let X be the base locus of a linear system L of hypersurfaces in PT (C). In this paper it is showed that the existence of linear syzygies for the ideal of X has strong consequences on the fibres of the rational map associated to L. The case of hyperquadrics is particularly addressed. The results are applied to the study of rational maps and to the Perazzo's map for cubic hypersurfaces.