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 | References | Similar Articles | Additional Information

Abstract: The *secant variety* of a projective variety in , denoted by , is defined to be the closure of the union of lines in passing through at least two points of , and the *secant deficiency* of is defined by . We list the homogeneous projective varieties with under the assumption that arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety with and , and the -variety is the only homogeneous projective variety with largest secant deficiency . 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.

**[B]**N. Bourbaki,*Éléments de Mathématique, Groupes et Algèbres de Lie, Chapitres 4,5 et 6*, Hermann, Paris, 1968. MR**39:1590****[F]**T. Fujita,*Projective threefolds with small secant varieties*, Sci. Papers College Gen. Ed. Univ. Tokyo**32**(1982), 33-46. MR**84d:14023****[FR]**T. Fujita, J. Roberts,*Varieties with small secant varieties: The extremal case*, Amer. J. Math.**103**(1981), 953-976. MR**82k:14042****[FH]**W. Fulton, J. Harris,*Representation Theory: A First Course*, Graduate Texts in Math.**129**, Springer-Verlag, New York, 1991. MR**93a:20069****[FL]**W. Fulton, R. Lazarsfeld,*Connectivity and its applications in algebraic geometry*, Algebraic Geometry, Lecture Notes in Math.**862**, Springer-Verlag, New York, 1981, pp. 26-92. MR**83i:14002****[Hr]**J. Harris,*Algebraic Geometry: A First Course*, Graduate Texts in Math.**133**, Springer-Verlag, New York, 1992. MR**93j:14001****[Ht]**R. Hartshorne,*Varieties with small codimension in projective space*, Bull. Amer. Math. Soc.**80 (6)**(1974), 1017-1032. MR**52:5688****[Hm]**J. E. Humphreys,*Introduction to Lie Algebras and Representation Theory*, Graduate Texts in Math.**9**, Springer-Verlag, New York, 1972. MR**48:2197****[KOY]**H. Kaji, M. Ohno, O. Yasukura,*Adjoint varieties and their secant varieties*, Indag. Math. (to appear).**[LV]**R. Lazarsfeld and A. Van de Ven,*Topics in the geometry of projective space. Recent work of F. L. Zak*, DMV Sem. 4, Birkhäuser Verlag, Basel and Boston, 1984. MR**87e:14045****[O]**M. Ohno,*On odd dimensional projective manifolds with smallest secant varieties*, Math. Z.**226**(1997), 483-498. CMP**98:05****[R]**J. Roberts,*Generic projections of algebraic varieties*, Amer. J. Math.**93**(1971), 191-214. MR**43:3263****[T]**H. Tango,*Remarks on varieties with small secant varieties*, Bull. Kyoto Univ. Ed., Ser. B**60**(1982), 1-10. MR**84d:14026****[Z]**F. L. Zak,*Tangents and Secants of Algebraic Varieties*, Translations of Math. Monographs, vol. 127, AMS, Providence, 1993. MR**94i:14053**

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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