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).