Finite abelian subgroups of the Cremona group of the plane. (English) Zbl 1111.14003

The Cremona group \(\text{Cr}(\mathbb{P}^2)\) of the plane is the group of birational maps of \(\mathbb{P}^2(\mathbb{C})\) into itself. The author presents the classification of finite cyclic subgroups of \(\text{Cr}(\mathbb{P}^2)\). He also gives some results that follow from this classification.


14E07 Birational automorphisms, Cremona group and generalizations
14E05 Rational and birational maps
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[1] Bayle, L.; Beauville, A., Birational involutions of \(\mathbb{P}^2\), Asian J. math., 4, 1, 11-17, (2000) · Zbl 1055.14012
[2] A. Beauville, p-elementary subgroups of the Cremona group, J. Algebra, in press · Zbl 1126.14017
[3] Beauville, A.; Blanc, J., On Cremona transformations of prime order, C. R. acad. sci. Paris, ser. I, 339, 257-259, (2004) · Zbl 1062.14017
[4] Blanc, J., Finite abelian subgroups of the Cremona group of the plane, Thesis, University of Geneva, 2006. Available online at
[5] G. Castelnuovo, Sulle transformazioni cremoniane del piano, che ammettono una curva fissa, Rend. Accad. Lincei (1892); Memorie scelte, Zanichelli, Bologna (1937)
[6] de Fernex, T., On planar Cremona maps of prime order, Nagoya math. J., 174, (2004) · Zbl 1062.14019
[7] I.V. Dolgachev, V.A. Iskovskikh, Finite subgroups of the plane Cremona group, in preparation · Zbl 1264.14017
[8] Gizatullin, M.K., Defining relations for the Cremona group of the plane, Izv. akad. nauk SSSR ser. mat., 46, 5, 909-970, (1982), 1134 · Zbl 0509.14011
[9] Iskovskikh, V.A., Minimal models of rational surfaces over arbitrary fields, Izv. akad. nauk SSSR ser. mat., 43, 1, 19-43, (1979), 237
[10] Iskovskikh, V.A., Generators and relations in a two-dimensional Cremona group, Vestnik moskov. univ. ser. I mat. mekh., 5, 310, 43-48, (1983) · Zbl 0522.14005
[11] Iskovskikh, V.A., Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Uspekhi mat. nauk, 51 4, 310, 3-72, (1996)
[12] Kantor, S., Theorie der endlichen gruppen von eindeutigen transformationen in der ebene, (1895), Mayer & Müller Berlin · JFM 26.0770.03
[13] Manin, Yu., Rational surfaces over perfect fields, II, Math. USSR-sb., 1, 141-168, (1967) · Zbl 0182.23701
[14] Wiman, A., Zur theorie der endlichen gruppen von birationalen transformationen in der ebene, Math. ann., 48, 497-498, (1896), 195-241
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