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

MathSciNet review:
1621761

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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. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines*, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR**0240238****[F]**Takao Fujita,*Projective threefolds with small secant varieties*, Sci. Papers College Gen. Ed. Univ. Tokyo**32**(1982), no. 1, 33–46. MR**674447****[FR]**Takao Fujita and Joel Roberts,*Varieties with small secant varieties: the extremal case*, Amer. J. Math.**103**(1981), no. 5, 953–976. MR**630774**, 10.2307/2374254**[FH]**William Fulton and Joe Harris,*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249****[FL]**William Fulton and Robert Lazarsfeld,*Connectivity and its applications in algebraic geometry*, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 26–92. MR**644817****[Hr]**Joe Harris,*Algebraic geometry*, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR**1182558****[Ht]**Robin Hartshorne,*Varieties of small codimension in projective space*, Bull. Amer. Math. Soc.**80**(1974), 1017–1032. MR**0384816**, 10.1090/S0002-9904-1974-13612-8**[Hm]**James E. Humphreys,*Introduction to Lie algebras and representation theory*, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR**0323842****[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*, DMV Seminar, vol. 4, Birkhäuser Verlag, Basel, 1984. Recent work of F. L. Zak; With an addendum by Zak. MR**808175****[O]**M. Ohno,*On odd dimensional projective manifolds with smallest secant varieties*, Math. Z.**226**(1997), 483-498. CMP**98:05****[R]**Joel Roberts,*Generic projections of algebraic varieties*, Amer. J. Math.**93**(1971), 191–214. MR**0277530****[T]**Hiroshi Tango,*Remark on varieties with small secant varieties*, Bull. Kyoto Univ. Ed. Ser. B**60**(1982), 1–10. MR**670136****[Z]**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**

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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