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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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