Automorphisms of $\mathbb {A}^{1}$-fibered affine surfaces
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- by Jérémy Blanc and Adrien Dubouloz PDF
- Trans. Amer. Math. Soc. 363 (2011), 5887-5924 Request permission
Abstract:
We develop techniques of birational geometry to study automorphisms of affine surfaces admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms and these fibrations. In particular, we associate to each surface $S$ of this type a graph encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the automorphism group of $S$ is generated by automorphisms preserving these fibrations.References
- Shreeram S. Abhyankar and Tzuong Tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166. MR 379502
- T. Bandman and L. Makar-Limanov, Affine surfaces with $\textrm {AK}(S)=\Bbb C$, Michigan Math. J. 49 (2001), no. 3, 567–582. MR 1872757, DOI 10.1307/mmj/1012409971
- Daniel Daigle, On locally nilpotent derivations of $k[X_1,X_2,Y]/(\phi (Y)-X_1X_2)$, J. Pure Appl. Algebra 181 (2003), no. 2-3, 181–208. MR 1975298, DOI 10.1016/S0022-4049(02)00334-1
- Adrien Dubouloz, Completions of normal affine surfaces with a trivial Makar-Limanov invariant, Michigan Math. J. 52 (2004), no. 2, 289–308. MR 2069802, DOI 10.1307/mmj/1091112077
- A. Dubouloz, Embeddings of Danielewski surfaces in affine spaces, Comment. Math. Helv. 81 (2006), no. 1, 49–73. MR 2208797, DOI 10.4171/CMH/42
- M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. I, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 3, 523–565, 703 (Russian). MR 0376701
- M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. II, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 1, 54–103, 231 (Russian). MR 0437545
- Javier Fernández de Bobadilla, A new geometric proof of Jung’s theorem on factorisation of automorphisms of ${\Bbb C}^2$, Proc. Amer. Math. Soc. 133 (2005), no. 1, 15–19. MR 2085147, DOI 10.1090/S0002-9939-04-07637-3
- Hubert Flenner, Shulim Kaliman, and Mikhail Zaidenberg, Birational transformations of weighted graphs, Affine algebraic geometry, Osaka Univ. Press, Osaka, 2007, pp. 107–147. MR 2327237
- M. H. Gizatullin, Quasihomogeneous affine surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1047–1071 (Russian). MR 0286791
- Heinrich W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161–174 (German). MR 8915, DOI 10.1515/crll.1942.184.161
- Stéphane Lamy, Une preuve géométrique du théorème de Jung, Enseign. Math. (2) 48 (2002), no. 3-4, 291–315 (French, with French summary). MR 1955604
- Stéphane Lamy, Sur la structure du groupe d’automorphismes de certaines surfaces affines, Publ. Mat. 49 (2005), no. 1, 3–20 (French, with English summary). MR 2140198, DOI 10.5565/PUBLMAT_{4}9105_{0}1
- Masayoshi Miyanishi, Open algebraic surfaces, CRM Monograph Series, vol. 12, American Mathematical Society, Providence, RI, 2001. MR 1800276, DOI 10.1090/crmm/012
- L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. Math. 69 (1990), no. 2, 250–256. MR 1045377, DOI 10.1007/BF02937308
- Jean-Pierre Serre, Arbres, amalgames, $\textrm {SL}_{2}$, Astérisque, No. 46, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass. MR 0476875
- W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3) 1 (1953), 33–41. MR 54574
Additional Information
- Jérémy Blanc
- Affiliation: Section de mathématiques, Université de Genève, 2-4 rue du Lièvre Case postale 64, 1211 Genève 4, Suisse
- Address at time of publication: Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Suisse
- MR Author ID: 744287
- Email: Jeremy.Blanc@unige.ch, Jeremy.Blanc@unibas.ch
- Adrien Dubouloz
- Affiliation: Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary - BP 47870, 21078 Dijon cedex, France
- Email: Adrien.Dubouloz@u-bourgogne.fr
- Received by editor(s): July 4, 2009
- Received by editor(s) in revised form: November 29, 2009
- Published electronically: June 1, 2011
- Additional Notes: This research has been partially supported by FABER Grant 07-512-AA-010-S-179
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 5887-5924
- MSC (2010): Primary 14R25, 14R20, 14R05, 14E05
- DOI: https://doi.org/10.1090/S0002-9947-2011-05266-9
- MathSciNet review: 2817414