On quasi-complete intersections of codimension 2

Abstract

In this paper, we prove that if X subset of P-n, n >= 4, is a locally complete intersection of pure codimension 2 and defined scheme-theoretically by three hypersurfaces of degrees d(1) >= d(2) >= d(3), then H-1(P-n, I-X(j)) = 0 for j < d(3) using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold X subset of P-5 is projectively normal if X is defined by three quintic hypersurfaces.