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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
##### Keywords:
rationality of quotient varieties; invariant mapping
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