Special linear systems and syzygies

Let $X$ be the base locus of a linear system $L$ of hypersurfaces in $\mathbb{P}^r(C)$. 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.