Projective subvarieties having large Green-Lazarsfeld index

Abstract

Let X subset of P(n+c) be a nondegenerate projective irreducible subvariety of degree d and codimension c >= 1. The Green-Lazarsfeld index of X, denoted by index(X), is defined as the largest p such that the homogeneous ideal of X is generated by quadrics and the syzygies among them are generated by linear syzygies until the (p - 1)-th stage. Thus index(X) is an important invariant in order to describe the minimal free resolution of X. Recently it is shown that d = c + 1 if and only if index(X) >= c, and X is a del Pezzo variety if and only if index(X) = c - 1. In this paper, we prove that index(X) = c - 2 (c >= 3) if and only if X is either a complete intersection of three quadrics or else an arithmetically Cohen-Macaulay variety with d = c + 3 (Theorem 1.1). Also we classify X with index(X) = c - 3 (c >= 4) for the cases when d = c + 2 (Theorem 4.1) and when X is a smooth surface (Theorem 4.3). (C) 2011 Elsevier Inc. All rights reserved.