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Rationality of some quotient varieties. (English. Russian original) Zbl 0591.14040
Math. USSR, Sb. 54, 571-576 (1986); translation from Mat. Sb., Nov. Ser. 126(168), No. 4, 584-589 (1985).
The authors establish the rationality of some varieties of type \({\mathbb{C}}^ n/G\) where G is an algebraic group which acts linearly on \({\mathbb{C}}^ n.\)
Main results:
Theorem 1. Let V be a linear space and G be one of the following groups: \(W_ n\) (the Weyl group) or \(PSL_ 3\), \(SL_ n\), \(SO_ n\), \(Sp_ n\). Suppose G acts on V linearly and locally free, then the field \({\mathbb{C}}(V)^ G(z_ 1,...,z_ m)\) is rational and m is equal respectively to n, 16, \(n^ 2, n^ 2, n^ 2\).
Theorem 2. Let V be a linear space and G an algebraic group acting on V linearly and locally free. Suppose that there exist linear representations G:W and linear locally free ones G:U such that (i) dim U- dim W-1, dim \(V\geq \dim U\), (ii) there exists a bilinear G-invariant mapping \(\psi:\quad V\times U\to W\) such that \(\psi (v_ 1,u_ 1)=0\), \(\psi (v_ 1,U)=\psi (V,u_ 1)=W\) for some point \((v_ 1,u_ 1)\in V\times U\). Then the fields \(({\mathbb{C}}(V)^ G)^{{\mathbb{C}}^*}\) and \(({\mathbb{C}}(U)^ G)^{{\mathbb{C}}^*}(z_ 1,...,z_ m)\) are isomorphic.
Theorem 3. Let V(2d) be a space of forms of degree 2d with respect to variables \(z_ 1,z_ 2\) and \(PSL_ 2:V(2d)\) the canonical action, then the field \(({\mathbb{C}}(V(2d)^{PSL_ 2})^{{\mathbb{C}}^*}\) is rational.
Reviewer: V.Janchevskij

MSC:
14M20 Rational and unirational varieties
14M17 Homogeneous spaces and generalizations
20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
14L24 Geometric invariant theory
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