The additive group actions on -homology planes
[Actions du groupe additif sur des plans -acycliques]
Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 429-464.

Dans cet article, on démontre qu’un plan -acyclique X avec deux actions du groupe additif G a qui sont algébriquement indépendantes, est isomorphe au plan affine ou bien au quotient d’une hypersurface affine xy=z n -1 dans l’espace affine de dimension 3 par une action de /m, où m est l’ordre d’un groupe fini H 1 (X;)

In this article, we prove that a -homology plane X with two algebraically independent G a -actions is isomorphic to either the affine plane or a quotient of an affine hypersurface xy=z m -1 in the affine 3-space via a free /m-action, where m is the order of a finite group H 1 (X;).

DOI : 10.5802/aif.1949
Classification : 14L30, 14R20, 14J26
Keywords: ${\mathbb {Q}}$-homology plane, additive group action, Makar-Limanov invariant
Mot clés : plan ${\mathbb {Q}}$-acyclique, action du groupe additif, invariant de Makar-Limanov
Masuda, Kayo 1 ; Miyanishi, Masayoshi 

1 Himeji Institute of Technology, Mathematical Sciences II, 2167 Shosha, Himeji 671-2201 (Japon)
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Masuda, Kayo; Miyanishi, Masayoshi. The additive group actions on ${\mathbb {Q}}$-homology planes. Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 429-464. doi : 10.5802/aif.1949. http://www.numdam.org/articles/10.5802/aif.1949/

[3] S. Bundagaard; J. Nielsen On normal subgroups with finite index in F-groups, Math. Tidsskrift, Volume B (1951), pp. 56-98 | MR | Zbl

[1] T. Bandman; L. Makar-Limanov Affine surfaces with A K ( S ) = , Michigan J. Math, Volume 49 (2001), pp. 567-582 | DOI | MR | Zbl

[2] J. Bertin Pinceaux de droites et automorphismes des surfaces affines, J. reine. angew. Math, Volume 341 (1983), pp. 32-53 | DOI | MR | Zbl

[4] K.-H. Fieseler On complex affine surfaces with + -action, Comment. Math. Helvetici, Volume 69 (1994), pp. 5-27 | DOI | MR | Zbl

[5] R. H. Fox On Fenchel's conjecture about F-groups, Math. Tidsskrift, Volume B (1952), pp. 61-65 | MR | Zbl

[6] R.V. Gurjar; M. Miyanishi On the Jacobian conjecture for -homology planes, J. reine angew. Math, Volume 516 (1999), pp. 115-132 | DOI | MR | Zbl

[7] R.V. Gurjar; C.R. Pradeep -homology planes are rational. III, Osaka J. Math, Volume 36 (1999) no. 2, pp. 259-335 | MR | Zbl

[8] S. Kaliman; L. Makar-Limanov On the Russell-Koras contractible threefolds. J. Algebraic Geom, J. Algebraic Geom, Volume 6 (1997) no. 2, pp. 247-268 | MR | Zbl

[9] K. Masuda; M. Miyanishi Étale endomorphisms of algebraic surfaces with G m -actions, Math. Ann, Volume 319 (2001), pp. 493-516 | DOI | MR | Zbl

[10] M. Miyanishi Curves on rational and unirational surfaces, Lecture Notes at Tata Institute of Fundamental Research, Springer, 1978 | MR | Zbl

[11] M. Miyanishi; K. Masuda Generalized Jacobian conjecture and related topics, Proceedings of the International Colloquium on Algebra, Arithmetic and Geometry, Mumbai 2000, Tata Institute of Fundamental Research (2002) | Zbl

[12] M. Miyanishi; T. Sugie Homology planes with quotient singularities, J. Math. Kyoto Univ, Volume 31 (1991), pp. 755-788 | MR | Zbl

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