On a classification of secant defective manifolds near the extremal case

Details On a classification of secant defective manifolds near the extremal case On a classification of secant defective manifolds near the extremal case

2011-08-22

arxiv.org/abs/1108.4264v1

Mathematics

Algebraic Geometry

9 pages

Scientific paper

Let $X\subset \mathbb{P}^N$ be a nondegenerate irreducible closed subvariety of dimension $n$ over the field of complex numbers and $SX$ be its secant variety in $\mathbb{P}^N$. $X$ is called `secant defective' if $\dim(SX)$ is strictly less than the expected dimension, $\min\{N,2n+1\}$. In \cite{Z1}, F.L. Zak showed that $N$ must be less than or equal to $M(n):={n+2 \choose n}-1$ if $X\subset\mathbb{P}^N$ is a secant defective manifold $(n\ge2)$ and that the Veronese variety $v_2(\mathbb{P}^n)$ is the only boundary case. R. Mu$\tilde{\textrm{n}}$oz, J. C. Sierra, and L. S. Conde classified next to extremal cases in \cite{MSC}. In this paper, we will consider the secant defective manifolds $X\subset\mathbb{P}^N$ of dimension $n$ with $N=M(n)-\epsilon$ for $\epsilon\ge0$. First, we will prove that $X$ is `conic-connected' for some small $\epsilon\ge 0$ (see Theorem 2.1) by showing that general tangential behavior of $X$ is good enough to apply Scorza lemma. Next, we will give classifications of secant defective manifolds for the range of $\epsilon\le n-2$ using a classification of conic-connected manifolds given by \cite{IR1}. Our method recovers previous results of \cite{Z1,MSC} and simplifies the proofs considerably using second fundamental form. Finally, we will end the paper by presenting an example of new secant defective manifolds near the extremal range which did not appear in \cite{MSC} (see Example 2.4).

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