Laudal-type theorems in P N

Abstract

In this paper we classify all integral, non-degenerate, locally Cohen-Macaulay subvarieties in P N, whose general complementary section is a complete intersection set of points: they are either complete intersections or curves on a quadric surface in P 3 or degree 4 arithmetically Buchsbaum surfaces in P 4 (i.e. the Veronese surface or a degeneration of it). As a consequence we show that every locally Cohen-Macaulay threefold in P S of degree 4 is a complete intersection. Moreover, we obtain a generalization of Laudal's Lemma to threefolds in P 5 and fourfolds in P 6, which gives a bound on the degree of a codimension 2, integral subvariety X in P N, depending both on N and a non-lifting level s of X.