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Rationality of fields of invariants. (English) Zbl 0659.14009
Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, No. 2, 3-16 (1987).
[For the entire collection see Zbl 0626.00011.]
This is a survey of numerous results obtained so far on the rationality or stable rationality of the various function fields arising from moduli problems in algebraic geometry.
Let G be an affine algebraic group acting linearly on a complex vector space V of finite dimension. A central problem discussed in this article is whether or not the field of G-invariants \({\mathbb{C}}(V)^ G\) is rational over \({\mathbb{C}}\). The author summarizes the known positive (sometimes negative) results by means of the methods used in the proofs. Then he turns to discussions of the rationality of the moduli spaces which are birationally equivalent to the orbit spaces of the type \({\mathbb{P}}(V)/G\).
Reviewer: M.Miyanishi

MSC:
14L24 Geometric invariant theory
14M20 Rational and unirational varieties
14D20 Algebraic moduli problems, moduli of vector bundles
14G05 Rational points