Homogeneous projective varieties with degenerate secants
HTML articles powered by AMS MathViewer
- by Hajime Kaji PDF
- Trans. Amer. Math. Soc. 351 (1999), 533-545 Request permission
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.References
- 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 [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- Takao Fujita, Projective threefolds with small secant varieties, Sci. Papers College Gen. Ed. Univ. Tokyo 32 (1982), no. 1, 33–46. MR 674447
- Takao Fujita and Joel Roberts, Varieties with small secant varieties: the extremal case, Amer. J. Math. 103 (1981), no. 5, 953–976. MR 630774, DOI 10.2307/2374254
- 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, DOI 10.1007/978-1-4612-0979-9
- 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
- Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
- Robin Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032. MR 384816, DOI 10.1090/S0002-9904-1974-13612-8
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- H. Kaji, M. Ohno, O. Yasukura, Adjoint varieties and their secant varieties, Indag. Math. (to appear).
- 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, DOI 10.1007/978-3-0348-9348-0
- M. Ohno, On odd dimensional projective manifolds with smallest secant varieties, Math. Z. 226 (1997), 483–498.
- Joel Roberts, Generic projections of algebraic varieties, Amer. J. Math. 93 (1971), 191–214. MR 277530, DOI 10.2307/2373457
- Hiroshi Tango, Remark on varieties with small secant varieties, Bull. Kyoto Univ. Ed. Ser. B 60 (1982), 1–10. MR 670136
- 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, DOI 10.1090/mmono/127
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
- Received by editor(s): April 9, 1996
- © Copyright 1999 American Mathematical Society
- 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
Dedicated: Dedicated to Professor Satoshi Arima on the occasion of his 70th birthday