Vanishing theorems, a theorem of Severi, and the equations defining projective varieties

In 1903 Severi [S] proved that if X, X' c P" are smooth surfaces whose union is the complete intersection of r 2 hypersurfaces of degrees di ... , dr-2, then hypersurfaces of degree k > Z di r cut out a complete linear series on X (cf. [SR, XIII.9.8]). The purpose of this paper is first of all to show that elementary arguments using the Kodaira vanishing theorem lead to a simple variant of Severi's statement (Theorem 7 below) which extends it in serval directions. More importantly, we hope to convince the reader that this result has a surprising number of applications to questions involving the equations defining projective varieties. Consider to begin with a smooth complex projective variety X C Pr of dimension n and codimension e = r n. In this setting, our theorem asserts the vanishing of the higher cohomology of suitable twists of the ath power of the ideal sheaf .Yx of X in Apr. Proposition 1. Assume that X is cut out scheme-theoretically in P" by hypersurfaces of degrees d1 ? d2> > d_. Then H (P'r,(k)) = 0 for i > 1 provided k > ad, + d2+ + de r. Note that only the degrees of the first e = codim(X, Pr) defining equations come into play here. When n = 2 and a = 1 this is a consequence of Severi's

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