CLASSIFICATION OF SECANT DEFECTIVE MANIFOLDS NEAR THE EXTREMAL CASE

Abstract

Let X subset of P-N be a nondegenerate irreducible closed subvariety of dimension n over the field of complex numbers and let SX subset of P-N be its secant variety. X subset of P-N is called 'secant defective' if dim(SX) is strictly less than the expected dimension 2n + 1. In a 1993 paper, F.L. Zak showed that for a secant defective manifold it is necessary that N <= ((n+2)(n)) - 1 and that the Veronese variety v(2)(P-n) is the only boundary case. Recently R. Munoz, J. C. Sierra, and L. E. Sola Conde classified secant defective varieties next to this extremal case.

In this paper, we will consider secant defective manifolds X subset of P-N of dimension n with N = ((n+2)(n)) - 1 - epsilon for epsilon >= 0. First, we will prove that X is an LQEL-manifold of type delta = 1 for epsilon <= n - 2 by showing that the tangential behavior of X is good enough to apply the Scorza lemma. Then we will completely describe the above manifolds by using the classification of conicconnected manifolds given by Ionescu and Russo. Our method generalizes previous results by Zak, and by Mu.noz, Sierra, and Sola Conde.