Classification of secant defective manifolds near the extremal case

Abstract:

Let $X\subset \mathbb {P}^N$ be a nondegenerate irreducible closed subvariety of dimension $n$ over the field of complex numbers and let $SX\subset \mathbb {P}^N$ be its secant variety. $X\subset \mathbb {P}^N$ 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 $N\le {n+2 \choose n}-1$ and that the Veronese variety $v_2(\mathbb {P}^n)$ is the only boundary case. Recently R. Muñoz, J. C. Sierra, and L. E. Solá Conde classified secant defective varieties next to this extremal case.

In this paper, we will consider secant defective manifolds $X\subset \mathbb {P}^N$ of dimension $n$ with $N={n+2 \choose n}-1-\epsilon$ for $\epsilon \ge 0$. First, we will prove that $X$ is an $LQEL$-manifold of type $\delta =1$ for $\epsilon \le n-2$ 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 conic-connected manifolds given by Ionescu and Russo. Our method generalizes previous results by Zak, and by Muñoz, Sierra, and Solá Conde.