Nagoya Mathematical Journal

Tangent loci and certain linear sections of adjoint varieties

Hajime Kaji and Osami Yasukura

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An adjoint variety $X(\mathfrak{g})$ associated to a complex simple Lie algebra $\mathfrak{g}$ is by definition a projective variety in $\mathbb {P}_{*}(\mathfrak{g})$ obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in $\mathfrak{g}$. We first describe the tangent loci of $X(\mathfrak{g})$ in terms of $\mathfrak{sl}_{2}$-triples. Secondly for a graded decomposition of contact type $\mathfrak{g} = \oplus_{-2 \le i \le 2}\, \mathfrak{g}_{i}$, we show that the intersection of $X(\mathfrak{g})$ and the linear subspace $\mathbb {P}_*(\mathfrak{g}_{1})$ in $\mathbb {P}_{*}$$(\mathfrak{g})$ coincides with the cubic Veronese variety associated to $\mathfrak{g}$.

Article information

Nagoya Math. J., Volume 158 (2000), 63-72.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14N05: Projective techniques [See also 51N35]
Secondary: 17B99: None of the above, but in this section


Kaji, Hajime; Yasukura, Osami. Tangent loci and certain linear sections of adjoint varieties. Nagoya Math. J. 158 (2000), 63--72.

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