We present some (unfortunately not all) known properties of the Cremona group; when it’s possible we mentioned links with the most known group of polynomial automorphisms of the affine plane. The mentioned properties are essentially algebraic properties: generators, relations, finite subgroups, subgroups of finite type, automorphisms of the Cremona group, Tits alternative... but also dynamical properties: classification of birational maps, centralizer, dynamic of an Heisenberg subgroup... We deal with a little the dynamical study of the iterates of a birational map and also with the construction of automorphisms with positive entropy.
On présente certaines (malheureusement pas toutes) propriétés connues du groupe de Cremona en faisant, lorsque c’est possible, un parallèle avec le groupe des automorphismes polynomiaux de . Les propriétés abordées seront essentiellement de nature algébrique : théorème de génération, sous-groupes finis, sous-groupes de type fini, description du groupe d’automorphismes du groupe de Cremona,... mais aussi de nature dynamique : classification des transformations birationnelles, centralisateur, dynamique d’un sous-groupe de Heisenberg... On évoque un peu les aspects concernant l’étude dynamique des itérés d’une transformation birationnelle ainsi que les problèmes de construction d’automorphismes de type entropique sur les surfaces rationnelles.
@article{TSG_2008-2009__27__45_0, author = {Julie D\'eserti}, title = {Quelques propri\'et\'es des transformations birationnelles du plan projectif complexe, une histoire pour {S.}}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {45--100}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {27}, year = {2008-2009}, doi = {10.5802/tsg.270}, language = {fr}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.270/} }
TY - JOUR AU - Julie Déserti TI - Quelques propriétés des transformations birationnelles du plan projectif complexe, une histoire pour S. JO - Séminaire de théorie spectrale et géométrie PY - 2008-2009 SP - 45 EP - 100 VL - 27 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.270/ DO - 10.5802/tsg.270 LA - fr ID - TSG_2008-2009__27__45_0 ER -
%0 Journal Article %A Julie Déserti %T Quelques propriétés des transformations birationnelles du plan projectif complexe, une histoire pour S. %J Séminaire de théorie spectrale et géométrie %D 2008-2009 %P 45-100 %V 27 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.270/ %R 10.5802/tsg.270 %G fr %F TSG_2008-2009__27__45_0
Julie Déserti. Quelques propriétés des transformations birationnelles du plan projectif complexe, une histoire pour S.. Séminaire de théorie spectrale et géométrie, Volume 27 (2008-2009), pp. 45-100. doi : 10.5802/tsg.270. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.270/
[1] M. Alberich-Carramiñana Plane Cremona maps, exceptional curves and roots, Trans. Amer. Math. Soc., Volume 357 (2005) no. 5, p. 1901-1914 (electronic) | MR | Zbl
[2] L. Bayle; A. Beauville Birational involutions of , Asian J. Math., Volume 4 (2000) no. 1, pp. 11-17 (Kodaira’s issue) | MR | Zbl
[3] A. Beauville Surfaces algébriques complexes, Société Mathématique de France, Paris, 1978 (Avec une sommaire en anglais, Astérisque, No. 54) | MR | Zbl
[4] A. Beauville -elementary subgroups of the Cremona group, J. Algebra, Volume 314 (2007) no. 2, pp. 553-564 | MR | Zbl
[5] A. Beauville; J. Blanc On Cremona transformations of prime order, C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 4, pp. 257-259 | MR | Zbl
[6] E. Bedford; K. Kim Periodicities in linear fractional recurrences : degree growth of birational surface maps, Michigan Math. J., Volume 54 (2006) no. 3, pp. 647-670 | MR | Zbl
[7] E. Bedford; K. Kim Dynamics of rational surface automorphisms : linear fractional recurrences, J. Geom. Anal., Volume 19 (2009) no. 3, pp. 553-583 | MR | Zbl
[8] E. Bedford; K. Kim Continuous Families of Rational Surface Automorphisms with Positive Entropy, Math. Ann., Volume 348 (2010) no. 3, pp. 667-688 | MR
[9] E. Bertini Ricerche sulle trasformazioni univoche involutorie nel piano, Annali di Mat., Volume 8 (1877), pp. 244-286
[10] G. Birkhoff Lie groups simply isomorphic with no linear group, Bull. Amer. Math. Soc., Volume 42 (1936) no. 12, pp. 883-888 | MR
[11] J. Blanc The number of conjugacy classes of elements of the Cremona group of some given finite order, Bull. Soc. Math. France, Volume 135 (2007) no. 3, pp. 419-434 | Numdam | MR | Zbl
[12] J. Blanc Elements and cyclic subgroups of finite order of the Cremona group (2008) (arXiv :0809.4673)
[13] J. Blanc On the inertia group of elliptic curves in the Cremona group of the plane, Michigan Math. J., Volume 56 (2008) no. 2, pp. 315-330 | MR | Zbl
[14] J. Blanc Linearisation of finite abelian subgroups of the Cremona group of the plane, Groups Geom. Dyn., Volume 3 (2009) no. 2, pp. 215-266 | MR | Zbl
[15] J. Blanc; I. Pan; T. Vust Sur un théorème de Castelnuovo, Bull. Braz. Math. Soc. (N.S.), Volume 39 (2008) no. 1, pp. 61-80 | MR | Zbl
[16] M. Brunella Birational geometry of foliations, Publicações Matemáticas do IMPA., Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2004 | MR | Zbl
[17] A. Campillo; J. Olivares Assigned base conditions and geometry of foliations on the projective plane, Singularities—Sapporo 1998 (Adv. Stud. Pure Math.), Volume 29, Kinokuniya, Tokyo, 2000, pp. 97-113 | MR | Zbl
[18] S. Cantat Dynamique des automorphismes des surfaces complexes compactes, École normale supérieure de Lyon (1999) (Masters thesis)
[19] S. Cantat Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad. Sci. Paris Sér. I Math., Volume 328 (1999) no. 10, pp. 901-906 | MR | Zbl
[20] S. Cantat Dynamique des automorphismes des surfaces , Acta Math., Volume 187 (2001) no. 1, pp. 1-57 | MR | Zbl
[21] S. Cantat Sur les groupes de transformations birationnelles des surfaces (2006) (preprint)
[22] S. Cantat; C. Favre Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math., Volume 561 (2003), pp. 199-235 | MR | Zbl
[23] S. Cantat; S. Lamy Groupes d’automorphismes polynomiaux du plan, Geom. Dedicata, Volume 123 (2006), pp. 201-221 | MR | Zbl
[24] S. Cantat; S. Lamy Normal subgroups in the Cremona group (2010) (arXiv :1007.0895)
[25] G. Castelnuovo Sulle transformazioni cremoniane del piano, che ammettono una curva fiera, Rend. Accad. Lincei, 1892
[26] G. Castelnuovo Le trasformationi generatrici del gruppo cremoniano nel piano, Atti della R. Accad. delle Scienze di Torino, Volume 36 (1901), pp. 861-874
[27] D. Cerveau; J. Déserti Transformations birationnelles de petit degré (2008) (arXiv :0811.2325, Accepté pour publication)
[28] V. I. Danilov Non-simplicity of the group of unimodular automorphisms of an affine plane, Mat. Zametki, Volume 15 (1974), pp. 289-293 | MR | Zbl
[29] T. de Fernex On planar Cremona maps of prime order, Nagoya Math. J., Volume 174 (2004), pp. 1-28 | MR | Zbl
[30] J. Déserti Groupe de Cremona et dynamique complexe : une approche de la conjecture de Zimmer, Int. Math. Res. Not. (2006), pp. Art. ID 71701, 27 | MR | Zbl
[31] J. Déserti Sur le groupe des automorphismes polynomiaux du plan affine, J. Algebra, Volume 297 (2006) no. 2, pp. 584-599 | MR | Zbl
[32] J. Déserti Sur les automorphismes du groupe de Cremona, Compos. Math., Volume 142 (2006) no. 6, pp. 1459-1478 | MR | Zbl
[33] J. Déserti Le groupe de Cremona est hopfien, C. R. Math. Acad. Sci. Paris, Volume 344 (2007) no. 3, pp. 153-156 | MR | Zbl
[34] J. Déserti Sur les sous-groupes nilpotents du groupe de Cremona, Bull. Braz. Math. Soc. (N.S.), Volume 38 (2007) no. 3, pp. 377-388 | MR
[35] J. Déserti Expériences sur certaines transformations birationnelles quadratiques, Nonlinearity, Volume 21 (2008) no. 6, pp. 1367-1383 | MR | Zbl
[36] J. A. Dieudonné La géométrie des groupes classiques, Springer-Verlag, Berlin, 1971 (Troisième édition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5) | MR | Zbl
[37] J. Diller; C. Favre Dynamics of bimeromorphic maps of surfaces, Amer. J. Math., Volume 123 (2001) no. 6, pp. 1135-1169 | MR | Zbl
[38] J. Diller; D. Jackson; A. Sommese Invariant curves for birational surface maps, Trans. Amer. Math. Soc., Volume 359 (2007) no. 6, p. 2793-2991 (electronic) | MR | Zbl
[39] I. V. Dolgachev; V. A. Iskovskikh Finite subgroups of the plane Cremona group, Algebra, arithmetic, and geometry : in honor of Yu. I. Manin. Vol. I (Progr. Math.), Volume 269, Birkhäuser Boston Inc., Boston, MA, 2009, pp. 443-548 | DOI | MR
[40] C. Favre Le groupe de Cremona et ses sous-groupes de type fini (À paraître dans Séminaire Bourbaki. Vol. 2008/2009, Exp. No. 998)
[41] G. Fischer Complex analytic geometry, Lecture Notes in Mathematics, Vol. 538, Springer-Verlag, Berlin, 1976 | MR | Zbl
[42] S. Friedland; J. Milnor Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynam. Systems, Volume 9 (1989) no. 1, pp. 67-99 | MR | Zbl
[43] J.-P. Furter; S. Lamy Normal subgroup generated by a plane polynomial automorphism (2009) (preprint) | MR
[44] É. Ghys Groups acting on the circle, Enseign. Math. (2), Volume 47 (2001) no. 3-4, pp. 329-407 | MR | Zbl
[45] M. Gizatullin On some tensor representations of the Cremona group of the projective plane, New trends in algebraic geometry (Warwick, 1996) (London Math. Soc. Lecture Note Ser.), Volume 264, Cambridge Univ. Press, Cambridge, 1999, pp. 111-150 | MR | Zbl
[46] M. H. Gizatullin Rational -surfaces, Izv. Akad. Nauk SSSR Ser. Mat., Volume 44 (1980) no. 1, p. 110-144, 239 | MR | Zbl
[47] M. H. Gizatullin The decomposition, inertia and ramification groups in birational geometry, Algebraic geometry and its applications (Yaroslavl, 1992) (Aspects Math., E25), Vieweg, Braunschweig, 1994, pp. 39-45 | MR | Zbl
[48] M. Kh. Gizatullin Defining relations for the Cremona group of the plane, Izv. Akad. Nauk SSSR Ser. Mat., Volume 46 (1982) no. 5, p. 909-970, 1134 | MR | Zbl
[49] X. Gómez-Mont; G. Kempf Stability of meromorphic vector fields in projective spaces, Comment. Math. Helv., Volume 64 (1989) no. 3, pp. 462-473 | MR | Zbl
[50] M. J. Greenberg Euclidean and non-Euclidean geometries, W. H. Freeman and Company, New York, 1993 (Development and history) | MR | Zbl
[51] V. A. Iskovskikh Minimal models of rational surfaces over arbitrary fields, Izv. Akad. Nauk SSSR Ser. Mat., Volume 43 (1979) no. 1, p. 19-43, 237 | MR | Zbl
[52] V. A. Iskovskikh Proof of a theorem on relations in the two-dimensional Cremona group, Uspekhi Mat. Nauk, Volume 40 (1985) no. 5(245), pp. 255-256 | MR | Zbl
[53] H. W. E. Jung Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., Volume 184 (1942), pp. 161-174 | MR
[54] S. Kantor Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Mayer & Müller, Berlin, 1895
[55] M. Koitabashi Automorphism groups of generic rational surfaces, J. Algebra, Volume 116 (1988) no. 1, pp. 130-142 | MR | Zbl
[56] J. Kollár; F. Mangolte Cremona transformations and diffeomorphisms of surfaces (2008) (arXiv :0809.3720) | MR
[57] S. Lamy L’alternative de Tits pour , J. Algebra, Volume 239 (2001) no. 2, pp. 413-437 | MR | Zbl
[58] S Lamy Une preuve géométrique du théorème de Jung, Enseign. Math. (2), Volume 48 (2002) no. 3-4, pp. 291-315 | MR | Zbl
[59] Ju. I. Manin Rational surfaces over perfect fields. II, Mat. Sb. (N.S.), Volume 72 (114) (1967), pp. 161-192 | MR | Zbl
[60] Yu. I. Manin Cubic forms, North-Holland Mathematical Library, 4, North-Holland Publishing Co., Amsterdam, 1986 (Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel) | MR | Zbl
[61] C. T. McMullen Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes Études Sci. (2007) no. 105, pp. 49-89 | Numdam | MR | Zbl
[62] M. McQuillan Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. (1998) no. 87, pp. 121-174 | Numdam | MR | Zbl
[63] L. G. Mendes Kodaira dimension of holomorphic singular foliations, Bol. Soc. Brasil. Mat. (N.S.), Volume 31 (2000) no. 2, pp. 127-143 | MR | Zbl
[64] M. Noether Ueber die auf Ebenen eindeutig abbildbaren algebraischen Flächen, Göttigen Nachr. (1869), pp. 1-6
[65] M. Noether Ueber Flächen, welche Schaaren rationaler Curven besitzen, Math. Ann., Volume 3(2) (1870), pp. 161-227 | MR
[66] M. Noether Zur Theorie der eindentigen Ebenentrasformationen, Math. Ann., Volume 5(4) (1872), pp. 635-639 | MR
[67] I. Pan Une remarque sur la génération du groupe de Cremona, Bol. Soc. Brasil. Mat. (N.S.), Volume 30 (1999) no. 1, pp. 95-98 | MR | Zbl
[68] I. Pan Sur le sous-groupe de décomposition d’une courbe irrationnelle dans le groupe de Cremona du plan, Michigan Math. J., Volume 55 (2007) no. 2, pp. 285-298 | MR | Zbl
[69] F. Ronga; T. Vust Birational diffeomorphisms of the real projective plane, Comment. Math. Helv., Volume 80 (2005) no. 3, pp. 517-540 | MR | Zbl
[70] K. Saito On a generalization of de-Rham lemma, Ann. Inst. Fourier (Grenoble), Volume 26 (1976) no. 2, pp. vii, 165-170 | Numdam | MR | Zbl
[71] P. E. Schupp Small cancellation theory over free products with amalgamation, Math. Ann., Volume 193 (1971), pp. 255-264 | MR | Zbl
[72] J. G. Semple; L. Roth Introduction to algebraic geometry, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1985 (Reprint of the 1949 original) | MR | Zbl
[73] J.-P. Serre Arbres, amalgames, , Société Mathématique de France, Paris, 1977 (Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46) | MR | Zbl
[74] R. Steinberg Some consequences of the elementary relations in , Finite groups—coming of age (Montreal, Que., 1982) (Contemp. Math.), Volume 45, Amer. Math. Soc., Providence, RI, 1985, pp. 335-350 | MR | Zbl
[75] J. Tits Free subgroups in linear groups, J. Algebra, Volume 20 (1972), pp. 250-270 | MR | Zbl
[76] A. Wiman Zur theorie der endlichen gruppen von birazionalen transformationen in der ebene, Math. Ann., Volume 48 (1896), pp. 195-240 | MR
[77] D. Wright Two-dimensional Cremona groups acting on simplicial complexes, Trans. Amer. Math. Soc., Volume 331 (1992) no. 1, pp. 281-300 | MR | Zbl
[78] R. J. Zimmer Actions of semisimple groups and discrete subgroups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) (1987), pp. 1247-1258 | MR | Zbl
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