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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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