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On quotient-varieties with an isolated singularity. (English. Russian original) Zbl 0987.14030
Sb. Math. 190, No. 4, 589-596 (1999); translation from Mat. Sb. 190, No. 4, 115-122 (1999).
Let \(k\) be an algebraically closed field of characteristic 0. Let \(G\) be a reductive algebraic linear group over \(k\) with \(\text{Lie} (G)\) simple and let \(\rho: G \to GL(V)\) be a finite dimensional rational representation. The classification of all pairs \((G, V)\) such that the quotient \(V// G\) has an isolated singularity is given. It is also shown that, for pairs \((G, V)\) with this property, the algebra of invariants \(k[V]^G\) has a natural \(\mathbb Z_+\)-grading such that \(\operatorname {Proj}k[V]^G\) is smooth and rational.
14L30 Group actions on varieties or schemes (quotients)
13A50 Actions of groups on commutative rings; invariant theory
20G15 Linear algebraic groups over arbitrary fields
14L24 Geometric invariant theory
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