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Rationality of the quotient of \(\mathbb P^2\) by finite group of automorphisms over arbitrary field of characteristic zero. (English) Zbl 1288.14009

Let \(k\) be a field of characteristic 0, and \(G\) be a finite group of automorphisms of the projective plane \(\mathbb P_{k}^2\) over \(k\), i.e., \(G\subset\text{PGL}_3(k)\). The main purpose of the article under review is to prove that the quotient variety \(\mathbb P_{k}^2/G\) is \(k\)-rational (i.e. purely transcendental over \(k\)). This result is an improvement of Castelnuovo’s rationality criterion, which requires that \(k\) is algebraically closed. As a corollary of the main result, the author obtains that if \(G\) is a finite subgroup of the general linear group \(\mathrm{GL}_3(k)\), then the field of invariants \(k(x_1,x_2,x_3)^G\) is \(k\)-rational.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
14E08 Rationality questions in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
14M20 Rational and unirational varieties
13A50 Actions of groups on commutative rings; invariant theory
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