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Tangent loci and certain linear sections of adjoint varieties. (English) Zbl 0982.14028
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 $${\mathfrak g}= \bigoplus_{-2\leq i\leq 2}$$ $${\mathfrak g}_i$$, 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 $${\mathfrak g}$$.

MSC:
 14N05 Projective techniques in algebraic geometry 17B20 Simple, semisimple, reductive (super)algebras 14M17 Homogeneous spaces and generalizations 17B70 Graded Lie (super)algebras
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