an:01803682
Zbl 1064.14041
Kaji, Hajime; Ohno, Masahiro; Yasukura, Osami
Adjoint varieties and their secant varieties
EN
Indag. Math., New Ser. 10, No. 1, 45-57 (1999).
00088543
1999
j
14J40 14M17 14N05
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.