On multisecants and the Green-Lazarsfeld index of projective varieties

Abstract

It is well known that a nondegenerate projective subvariety X ⊂ P r of degree d and codimension c > 1 has minimal degree (i.e., d = c + 1) if and only if index(X) ≥ c if and only if X has no multisecant c-space. In this paper we extend this result by classifying varieties with index(X) ≥ c − s or with no multisecant (c − s)-space for s = 1 and 2. Mathematics Subject Classification (2010) . Primary 14N05, 14N25.Keywords. Multisecant space, Green-Lazarsfeld index, Variety of low degree. Introduction.Let X ⊂ P r be a nondegenerate projective subvariety of degree d and codimension c > 1 over an algebraically closed field K of arbitrary characteristic. A k-dimensional linear subspace Λ in P r is called an -secant k-plane to X if dim X ∩ Λ = 0 and length(X ∩ Λ) ≥ . We say that Λ is a multisecant k-space to X if > k + 1. Recently, the authors in [6] found a nice connection between the multisecant spaces to X and the minimal free resolution of the homogeneous ideal I X of X. According to [6], X is said to satisfy the property N 2,p if I X is generated by quadrics and the syzygies among them are generated by linear syzygies until the (p − 1)-th step. Also due to [4], the largest such p is called the Green-Lazarsfeld index of X and denoted by index(X). A result in [6] shows that index(X) < p if X has a multisecant p-space. For example, if d > c + 1 then X has a multisecant c-space and so index(X) < c. This implies that the following are equivalent: