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Homogeneous projective varieties with degenerate secants. (English) Zbl 0905.14031
Summary: The secant variety of a projective variety \(X\) in \(\mathbb{P}\), denoted by \(\text{Sec }X\), 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 \(\delta := 2 \dim X + 1 - \dim \text{Sec }X\). We list the homogeneous projective varieties \(X\) with \(\delta > 0\) under the assumption that \(X\) arises from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety \(X\) with \(\text{Sec }X \not = \mathbb{P}\) 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 [“Topics in the geometry of projective spaces. Recent work of F. L. Zak”, DMV Seminar, Vol. 4 (1984; Zbl 0564.14007); §1f] if we restrict ourselves to homogeneous projective varieties.

MSC:
14M17 Homogeneous spaces and generalizations
14N05 Projective techniques in algebraic geometry
20G05 Representation theory for linear algebraic groups
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[1] 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). · Zbl 0186.33001
[2] Takao Fujita, Projective threefolds with small secant varieties, Sci. Papers College Gen. Ed. Univ. Tokyo 32 (1982), no. 1, 33 – 46. · Zbl 0492.14027
[3] Takao Fujita and Joel Roberts, Varieties with small secant varieties: the extremal case, Amer. J. Math. 103 (1981), no. 5, 953 – 976. · Zbl 0475.14046 · doi:10.2307/2374254 · doi.org
[4] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. · Zbl 0744.22001
[5] 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. · Zbl 0484.14005
[6] Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. · Zbl 0932.14001
[7] Robin Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017 – 1032. · Zbl 0304.14005
[8] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. · Zbl 0254.17004
[9] H. Kaji, M. Ohno, O. Yasukura, Adjoint varieties and their secant varieties, Indag. Math. (to appear). · Zbl 1064.14041
[10] 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. · Zbl 0564.14007
[11] M. Ohno, On odd dimensional projective manifolds with smallest secant varieties, Math. Z. 226 (1997), 483-498. CMP 98:05 · Zbl 0887.14029
[12] Joel Roberts, Generic projections of algebraic varieties, Amer. J. Math. 93 (1971), 191 – 214. · Zbl 0212.53801 · doi:10.2307/2373457 · doi.org
[13] Hiroshi Tango, Remark on varieties with small secant varieties, Bull. Kyoto Univ. Ed. Ser. B 60 (1982), 1 – 10. · Zbl 0505.14032
[14] 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. · Zbl 0795.14018
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