Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14M17, 14N05, 17B10, 20G05

Retrieve articles in all journals with MSC (1991): 14M17, 14N05, 17B10, 20G05

Additional Information

Hajime Kaji
Affiliation: Department of Mathematics School of Science and Engineering Waseda University 3-4-1 Ohkubo Shinjuku-ku Tokyo 169, Japan

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

American Mathematical Society