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Homogeneous projective varieties with
degenerate secants


Author: Hajime Kaji
Journal: Trans. Amer. Math. Soc. 351 (1999), 533-545
MSC (1991): Primary 14M17, 14N05, 17B10, 20G05
DOI: https://doi.org/10.1090/S0002-9947-99-02378-8
MathSciNet review: 1621761
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Abstract: The secant variety of a projective variety $X$ in $\mathbb{P}$, denoted by $\operatorname{Sec}X$, is defined to be the closure of the union of lines in $\mathbb{P}$ passing through at least two points of $X$, and the secant deficiency of $X$ is defined by $\delta := 2 \dim X + 1 - \dim \operatorname{Sec}X$. We list the homogeneous projective varieties $X$ with $\delta > 0$ under the assumption that $X$ arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety $X$ with $\operatorname{Sec}X \not = \mathbb{P}$ and $\delta > 8$, and the $E_{6}$-variety is the only homogeneous projective variety with largest secant deficiency $\delta = 8$. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.


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Additional Information

Hajime Kaji
Affiliation: Department of Mathematics School of Science and Engineering Waseda University 3-4-1 Ohkubo Shinjuku-ku Tokyo 169, Japan
Email: kaji@mse.waseda.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-99-02378-8
Received by editor(s): April 9, 1996
Dedicated: Dedicated to Professor Satoshi Arima on the occasion of his 70th birthday
Article copyright: © Copyright 1999 American Mathematical Society

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