Geodesic
Rational surfaces with a large group of automorphisms
Cantat, Serge1 ; Dolgachev, Igor2
1 IRMAR, UMR 6625 du CNRS et Université de Rennes 1, Bât. 22-23 du Campus de Beaulieu, F-35042 Rennes cedex; DMA, UMR 8553 du CNRS, École Normale Supérieure de Paris, 45 rue d’Ulm, F-75230 Paris cedex 05
2 Department of Mathematics, University of Michigan, 525 E. University Avenue, Ann Arbor, Michigan, 49109
Journal of the American Mathematical Society, Tome 25 (2012) no. 3, pp. 863-905

Voir la notice de l'article provenant de la source American Mathematical Society

We classify rational surfaces $X$ for which the image of the automorphism group $\mathrm {Aut}(X)$ in the group of linear transformations of the Picard group $\mathrm {Pic}(X)$ is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in terms of periodic orbits of birational actions of infinite Coxeter groups.
DOI : 10.1090/S0894-0347-2012-00732-2

Cantat, Serge 1 ; Dolgachev, Igor 2

1 IRMAR, UMR 6625 du CNRS et Université de Rennes 1, Bât. 22-23 du Campus de Beaulieu, F-35042 Rennes cedex; DMA, UMR 8553 du CNRS, École Normale Supérieure de Paris, 45 rue d’Ulm, F-75230 Paris cedex 05
2 Department of Mathematics, University of Michigan, 525 E. University Avenue, Ann Arbor, Michigan, 49109 
@article{10_1090_S0894_0347_2012_00732_2,
     author = {Cantat, Serge and Dolgachev, Igor},
     title = {Rational surfaces with a large group of automorphisms},
     journal = {Journal of the American Mathematical Society},
     pages = {863--905},
     publisher = {mathdoc},
     volume = {25},
     number = {3},
     year = {2012},
     doi = {10.1090/S0894-0347-2012-00732-2},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2012-00732-2/}
}
TY  - JOUR
AU  - Cantat, Serge
AU  - Dolgachev, Igor
TI  - Rational surfaces with a large group of automorphisms
JO  - Journal of the American Mathematical Society
PY  - 2012
SP  - 863
EP  - 905
VL  - 25
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2012-00732-2/
DO  - 10.1090/S0894-0347-2012-00732-2
ID  - 10_1090_S0894_0347_2012_00732_2
ER  - 
%0 Journal Article
%A Cantat, Serge
%A Dolgachev, Igor
%T Rational surfaces with a large group of automorphisms
%J Journal of the American Mathematical Society
%D 2012
%P 863-905
%V 25
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2012-00732-2/
%R 10.1090/S0894-0347-2012-00732-2
%F 10_1090_S0894_0347_2012_00732_2
Cantat, Serge; Dolgachev, Igor. Rational surfaces with a large group of automorphisms. Journal of the American Mathematical Society, Tome 25 (2012) no. 3, pp. 863-905. doi: 10.1090/S0894-0347-2012-00732-2

[1] Barth, Wolf P., Hulek, Klaus, Peters, Chris A. M., Van De Ven, Antonius Compact complex surfaces 2004

[2] Barth, W., Peters, C. Automorphisms of Enriques surfaces Invent. Math. 1983 383 411

[3] Bedford, Eric, Kim, Kyounghee Periodicities in linear fractional recurrences: degree growth of birational surface maps Michigan Math. J. 2006 647 670

[4] Benoist, Yves, De La Harpe, Pierre Adhérence de Zariski des groupes de Coxeter Compos. Math. 2004 1357 1366

[5] Cantat, Serge Dynamique des automorphismes des surfaces 𝐾3 Acta Math. 2001 1 57

[6] Cantat, S. Sur la dynamique du groupe d’automorphismes des surfaces 𝐾3 Transform. Groups 2001 201 214

[7] Cantat, Serge Bers and Hénon, Painlevé and Schrödinger Duke Math. J. 2009 411 460

[8] Cantat, Serge, Favre, Charles Symétries birationnelles des surfaces feuilletées J. Reine Angew. Math. 2003 199 235

[9] Coble, Arthur B. The Ten Nodes of the Rational Sextic and of the Cayley Symmetroid Amer. J. Math. 1919 243 265

[10] Coble, Arthur B. Algebraic geometry and theta functions 1982

[11] Cossec, F., Dolgachev, I. Smooth rational curves on Enriques surfaces Math. Ann. 1985 369 384

[12] Cossec, Franã§Ois R., Dolgachev, Igor V. Enriques surfaces. I 1989

[13] Séminaire sur les Singularités des Surfaces 1980

[14] Deodhar, Vinay V. A note on subgroups generated by reflections in Coxeter groups Arch. Math. (Basel) 1989 543 546

[15] Dolgachev, I. On automorphisms of Enriques surfaces Invent. Math. 1984 163 177

[16] Dolgachev, I. Infinite Coxeter groups and automorphisms of algebraic surfaces 1986 91 106

[17] Dolgachev, Igor V. Reflection groups in algebraic geometry Bull. Amer. Math. Soc. (N.S.) 2008 1 60

[18] Dolgachev, Igor, Ortland, David Point sets in projective spaces and theta functions Astérisque 1988

[19] Dolgachev, Igor V., Zhang, De-Qi Coble rational surfaces Amer. J. Math. 2001 79 114

[20] Douady, Adrien, Hubbard, John H. A proof of Thurston’s topological characterization of rational functions Acta Math. 1993 263 297

[21] Gizatullin, M. H. Rational 𝐺-surfaces Izv. Akad. Nauk SSSR Ser. Mat. 1980

[22] Harbourne, Brian Automorphisms of 𝐾3-like rational surfaces 1987 17 28

[23] Harbourne, Brian Rational surfaces with infinite automorphism group and no antipluricanonical curve Proc. Amer. Math. Soc. 1987 409 414

[24] Hartshorne, Robin Algebraic geometry 1977

[25] Kneser, Martin Quadratische Formen 2002

[26] Knus, Max-Albert Quadratic and Hermitian forms over rings 1991

[27] Mcmullen, Curtis T. Dynamics on blowups of the projective plane Publ. Math. Inst. Hautes Études Sci. 2007 49 89

[28] Morrison, D. R. On 𝐾3 surfaces with large Picard number Invent. Math. 1984 105 121

[29] Mumford, David Lectures on curves on an algebraic surface 1966

[30] Nagata, Masayoshi On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1 Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 1960 351 370

[31] Nagata, Masayoshi On rational surfaces. II Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 1960/61 271 293

[32] Nikulin, V. V. Integer symmetric bilinear forms and some of their geometric applications Izv. Akad. Nauk SSSR Ser. Mat. 1979

[33] Serre, Jean-Pierre Cours d’arithmétique 1977 188

[34] Takenawa, Tomoyuki Algebraic entropy and the space of initial values for discrete dynamical systems J. Phys. A 2001 10533 10545

Cité par Sources :