Projective subvarieties having large Green-Lazarsfeld index

Let X⊂Pn+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).