Let $$\Bbbk$$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $$\Bbbk$$. Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $$\Bbbk$$ is algebraically closed. In this paper we prove that $${{\mathbb{P}_\Bbbk ^2 } \mathord{\left/ {\vphantom {{\mathbb{P}_\Bbbk ^2 } G}} \right. \kern-\nulldelimiterspace} G}$$ is rational for an arbitrary field $$\Bbbk$$ of characteristic zero.
[1] Ahmad H., Hajja M., Kang M., Rationality of some projective linear actions, J. Algebra, 2000, 228(2), 643–658 http://dx.doi.org/10.1006/jabr.2000.8292
[2] Artebani M., Dolgachev I., The Hesse pencil of plane cubic curves, Enseign. Math., 2009, 55(3–4), 235–273
[3] Blichfeldt H.F., Finite Collineation Groups, University of Chicago Press, Chicago, 1917
[4] Bogomolov F.A., The Brauer group of quotient spaces by linear group actions, Math. USSR-Izv., 1988, 30(3), 455–485 http://dx.doi.org/10.1070/IM1988v030n03ABEH001024
[5] Bogomolov F.A., Katsylo P.I., Rationality of some quotient varieties, Mat. Sb. (N.S.), 1985, 126(168)(4), 584–589 (in Russian)
[6] Borel A., Linear Algebraic Groups, 2nd ed., Grad. Texts in Math., 126, Springer, New York, 1991 http://dx.doi.org/10.1007/978-1-4612-0941-6
[7] Coray D.F., Tsfasman M.A., Arithmetic on singular Del Pezzo surfaces, Proc. Lond. Math. Soc., 1988, 57(1), 25–87 http://dx.doi.org/10.1112/plms/s3-57.1.25
[8] Dolgachev I.V., Iskovskikh V.A., Finite subgroups of the plane Cremona group, In: Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin, I, Progr. Math., 269, Birkhäuser, Basel, 2009, 443–548 http://dx.doi.org/10.1007/978-0-8176-4745-2_11
[9] Dolgachev I.V., Iskovskikh V.A., On elements of prime order in the plane Cremona group over a perfect field, Int. Math. Res. Not. IMRN, 2009, 18, 3467–3485
[10] Endô S., Miyata T., Invariants of finite abelian groups, J. Math. Soc. Japan, 1973, 25, 7–26 http://dx.doi.org/10.2969/jmsj/02510007
[11] Hajja M., Rationality of finite groups of monomial automorphisms of k(x; y), J. Algebra, 1987, 109(1), 46–51 http://dx.doi.org/10.1016/0021-8693(87)90162-1
[12] Hoshi A., Kang M., Unramified Brauer groups for groups of order p5, preprint aviable at http://arxiv.org/abs/1109.2966
[13] Iskovskikh V.A., Minimal models of rational surfaces over arbitrary fields, Math. USSR-Izv., 1980, 14(1), 17–39 http://dx.doi.org/10.1070/IM1980v014n01ABEH001064
[14] Iskovskikh V.A., Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Russian Math. Surveys, 1996, 51(4), 585–652 http://dx.doi.org/10.1070/RM1996v051n04ABEH002962
[15] Lenstra H.W. Jr., Rational functions invariant under a finite abelian group, Invent. Math., 1974, 25(3–4), 299–325 http://dx.doi.org/10.1007/BF01389732
[16] Manin Ju.I., Rational surfaces over perfect fields II, Mat. Sb., 1967, 1(2), 141–168 http://dx.doi.org/10.1070/SM1967v001n02ABEH001972
[18] Miller G.A., Blichfeldt H.F., Dickson L.E., Theory and Applications of Finite Groups, Dover, New York, 1961
[19] Moravec P., Unramified Brauer groups of finite and infinite groups, Amer. J. Math., 2012, 134(6), 1679–1704 http://dx.doi.org/10.1353/ajm.2012.0046
[20] Noether E., Rationale Functionenkörper, Jahresbericht der Deutschen Mathematiker-Vereinigung, 1913, 22, 316–319
[21] Prokhorov Yu.G., Fields of invariants of finite linear groups, In: Cohomological and Geometric Approaches to Rationality Problems, Progr. Math., 282, Birkhäuser, Boston, 2010, 245–273 http://dx.doi.org/10.1007/978-0-8176-4934-0_10
[22] Saltman D.J., Noether’s problem over an algebraically closed field, Invent. Math., 1984, 77(1), 71–84 http://dx.doi.org/10.1007/BF01389135
[23] Shephard G.C., Todd J.A., Finite unitary reflection groups, Canadian J. Math., 1954, 6, 274–304 http://dx.doi.org/10.4153/CJM-1954-028-3
[24] Swan R.G., Invariant rational functions and a problem of Steenrod, Invent. Math., 1969, 7, 148–158 http://dx.doi.org/10.1007/BF01389798
[25] Voskresenskii V.E., On two-dimensional algebraic tori II, Math. USSR-Izv., 1967, 1(3), 691–696 http://dx.doi.org/10.1070/IM1967v001n03ABEH000580
[26] Voskresenskii V.E., Fields of invariants for abelian groups, Russian Math. Surveys, 1973, 28(4), 79–105 http://dx.doi.org/10.1070/RM1973v028n04ABEH001594