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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
DOI: https://doi.org/10.1090/S0002-9939-2013-11715-6
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.


References [Enhancements On Off] (What's this?)

  • [Ad87] Bjørn Ådlandsvik, Joins and higher secant varieties, Math. Scand. 61 (1987), no. 2, 213-222. MR 947474 (89j:14030)
  • [Ara06] Carolina Araujo, Rational curves of minimal degree and characterizations of projective spaces, Math. Ann. 335 (2006), no. 4, 937-951. MR 2232023 (2007d:14037), https://doi.org/10.1007/s00208-006-0775-2
  • [CC01] L. Chiantini and C. Ciliberto, Threefolds with degenerate secant variety: on a theorem of G. Scorza, Geometric and combinatorial aspects of commutative algebra (Messina, 1999), Lecture Notes in Pure and Appl. Math., vol. 217, Dekker, New York, 2001, pp. 111-124. MR 1824221 (2002d:14088)
  • [CMSB02] Koji Cho, Yoichi Miyaoka, and N. I. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997) Adv. Stud. Pure Math., vol. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1-88. MR 1929792 (2003m:14080)
  • [Fuj82] Takao Fujita, Projective threefolds with small secant varieties, Sci. Papers College Gen. Ed. Univ. Tokyo 32 (1982), no. 1, 33-46. MR 674447 (84d:14023)
  • [FR81] Takao Fujita and Joel Roberts, Varieties with small secant varieties: the extremal case, Amer. J. Math. 103 (1981), no. 5, 953-976. MR 630774 (82k:14042), https://doi.org/10.2307/2374254
  • [IR09] P. Ionescu and F. Russo, Manifolds covered by lines, defective manifolds and a restricted Hartshorne conjecture, preprint (arXiv:0909.2763).
  • [IR10] Paltin Ionescu and Francesco Russo, Conic-connected manifolds, J. Reine Angew. Math. 644 (2010), 145-157. MR 2671777 (2011f:14083), https://doi.org/10.1515/CRELLE.2010.054
  • [Mor79] Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593-606. MR 554387 (81j:14010), https://doi.org/10.2307/1971241
  • [MSS07] Roberto Muñoz, José Carlos Sierra, and Luis E. Solá Conde, Tangential projections and secant defective varieties, Bull. Lond. Math. Soc. 39 (2007), no. 6, 949-961. MR 2392818 (2009b:14098), https://doi.org/10.1112/blms/bdm098
  • [Rus08] F. Russo, Geometry of Special Varieties, preprint.
  • [Rus09] Francesco Russo, Varieties with quadratic entry locus. I, Math. Ann. 344 (2009), no. 3, 597-617. MR 2501303 (2010i:14092), https://doi.org/10.1007/s00208-008-0318-0
  • [Sco08] G. Scorza, Determinazione delle varietà a tre dimensioni di $ S_r$, $ r\ge 7$, i cui $ S_3$ tangenti si tagliano due a due, Rend. Circ. Mat. Palermo 25 (1908), 167-182.
  • [Sco09] G. Scorza, Sulle varietà a quattro dimensioni di $ S_r$, $ r\ge 9$, i cui $ S_4$ tangenti si tagliano a due a due, Rend. Circ. Mat. Palermo 27 (1909), 148-178.
  • [Sev01] F. Severi, Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni e ai suoi punti tripli apparenti, Rend. Circ. Mat. Palermo 15 (1901), 33-51.
  • [Ter11] A. Terracini, Sulle $ V_k$ per cui la varietà degli $ S_h$ $ (h+1)$-secanti ha dimensione minore dell' ordinario, Rend. Circ. Mat. Palermo 31 (1911), 392-396.
  • [Zak93] F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494 (94i:14053)

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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
Email: han.kangjin@kaist.ac.kr, kangjin.han@kias.re.kr

DOI: https://doi.org/10.1090/S0002-9939-2013-11715-6
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.

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