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Classification of secant defective manifolds near the extremal case

Author: Kangjin Han
Journal: Proc. Amer. Math. Soc. 142 (2014), 39-46
MSC (2010): Primary 14Mxx, 14Nxx, 14M22
Published electronically: September 10, 2013
MathSciNet review: 3119179
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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.

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Additional Information

Kangjin Han
Affiliation: Algebraic Structure and its Applications Research Center (ASARC), Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong, Yusung-Gu, Daejeon, Republic of Korea
Address at time of publication: School of Mathematics, Korean Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea

Keywords: Secant defective, local quadratic entry locus, conic-connected, Terracini lemma, tangential projection, second fundamental form, Scorza lemma.
Received by editor(s): August 31, 2011
Received by editor(s) in revised form: January 27, 2012, and February 23, 2012
Published electronically: September 10, 2013
Additional Notes: This work was supported by the National Research Foundation of Korea (NRF) with a grant funded by the Korean government (MEST) (No. 2011-0001182)
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.