## 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.

### MSC:

 1.4e+08 Birational automorphisms, Cremona group and generalizations 1.4e+06 Rational and birational maps

### Keywords:

Cremona group; birational maps
Full Text:

### References:

 [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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