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Adjoint varieties and their secant varieties. (English) Zbl 1064.14041
Summary: The purpose of this article is to show how the graded decomposition of complex simple Lie algebras can be applied to studying adjoint varieties $$X$$ and their secant varieties Sec$$(X)$$. Firstly quadratic equations defining adjoint varieties are explicitly given. Secondly it is shown that dim Sec$$(X) = 2~\text{dim} X$$ for adjoint varieties X in two ways: one is based on Terracini’s lemma, and the other is on some explicit description of Sec$$(X)$$ in terms of an orbit of the adjoint action. Finally it is shown that the contact loci of the secant variety to its embedded tangent space have dimension two if $$X$$ is adjoint.

##### MSC:
 14J40 $$n$$-folds ($$n>4$$) 14M17 Homogeneous spaces and generalizations 14N05 Projective techniques in algebraic geometry
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##### References:
 [1] Asano, H., On triple systems, Yokohama city univ. ronso. ser. natural sci., 27, 7-31, (1975), (in Japanese) [2] Asano, H., Symplectic triple systems and simple Lie algebras, RIMS kokyuroku. Kyoto univ., 308, 41-54, (1977), (in Japanese) [3] Boothby, W., Homogeneous complex contact manifolds, (), 144-154 [4] Boothby, W., A note on homogeneous complex contact manifolds, (), 276-280 · Zbl 0103.38703 [5] Bourbaki, N., Éléments de mathématique, groupes et algèbres de Lie, (1968), Hermann Paris, Chapitres 4, 5 et 6 · Zbl 0186.33001 [6] Dynkin, E.B.; Dynkin, E.B., Semisimple subalgebras of semisimple Lie algebras, Mat. sbornik N.S., AMS translations series 2, 6, 72, 111-244, (1957), (English translations) · Zbl 0077.03404 [7] Freudenthal, H. — Beziehungen der E7 und E8 zur Oktavenebene. VIII; IX. Nederl. Akad. Wetensch. Proc. Ser. A 62 [8] Freudenthal, H. — Beziehungen der E7 und E8 zur Oktavenebene, X. Nederl. Akad. Wetensch. Proc. Ser. A 66 [9] Fujita, T.; Roberts, J., Varieties with small secant varieties: the extremal case, Amer. J. math., 103, 953-976, (1981) · Zbl 0475.14046 [10] Fujita, T., Projective threefolds with small secant varieties, Sci. papers college gen. ed. univ. Tokyo, 32, 33-46, (1982) · Zbl 0492.14027 [11] Fulton, W.; Harris, J., Representation theory: A first course, () · Zbl 0744.22001 [12] Harris, J., Algebraic geometry: A first course, () · Zbl 0779.14001 [13] Humphreys, J.E., Introduction to Lie algebras and representation theory, () · Zbl 0254.17004 [14] Humphreys, J.E., Linear algebraic groups, () · Zbl 0507.20017 [15] Kaji, H. — Homogeneous projective varieties with degenerate secants. (To appear in Trans. Amer. Math. Soc.). · Zbl 0905.14031 [16] Kaji, H. and O. Yasukura — Adjoint varieties and their secant varieties. II. (In preparation). · Zbl 1064.14041 [17] Lazarsfeld, R.; van de Ven, A., Topics in the geometry of projective space, () · Zbl 0564.14007 [18] Lichtenstein, W., A system of quadrics describing the orbit of the highest weight vector, (), 605-608 · Zbl 0501.22017 [19] Ohno, M., On odd dimensional projective manifolds with smallest secant varieties, Math. Z., 226, 483-498, (1977) · Zbl 0887.14029 [20] Ohno, M. — On degenerate secant varieties whose Gauss maps have the largest images, (To appear in Pacific J. Math.). · Zbl 0939.14029 [21] Wolf, J.A., Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. math. and mechanics, 14, 1033-1047, (1965) · Zbl 0141.38202 [22] Yamaguti, K., On metasymplectic geometry, RIMS kokyuroku, Kyoto univ., 308, 55-92, (1977), (in Japanese) [23] Yamaguti, K.; Asano, H., On Freudenthal’s construction of exceptional Lie algebras, (), 253-258 · Zbl 0298.17010 [24] Yasukura, O., On subalgebras of type A1 in simple Lie algebras, Algebras, groups and geometries, 5, 359-368, (1988) [25] Zak, F.L., Tangents and secants of algebraic varieties, () · Zbl 0795.14018
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