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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.

If X is a finite set of points in a multiprojective space P^n1 x ... x P^nr with r >= 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For sets of points in P^1 x P^1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space. We show that each classification for ACM points in P^1 x P^1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM., 21 pages; revised final version; minor corrections; to appear in Collectanea Mathematica