Israel Journal of Mathematics

, Volume 121, Issue 1, pp 113–123

On the group of automorphisms of a surfacexny=P(z)

  • L. Makar-Limanov
Article

Abstract

In this note the AK invariant of a surface in ℂ3 which is given byxny=P(z) wheren>1 and deg(P)=d>1 is computed. Then this information is used to find the group of automorphisms of this surface and the isomorphism classes of such surfaces.

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References

  1. [AEH] S. Abhyankar, P. Eakin and W. Heinzer,On the uniqueness of the coefficient ring in a polynomial ring, Journal of Algebra23 (1972), 310–342.MATHCrossRefMathSciNetGoogle Scholar
  2. [B] J. Bertin,Pinceaux de droites et automorphismes des surfaces affines, Journal für die Reine und Angewandte Mathematik341 (1983), 32–53.MATHMathSciNetGoogle Scholar
  3. [C] P. M. Cohn,Free Rings and their Relations, second edition, Academic Press, New York, 1985.MATHGoogle Scholar
  4. [Da] W. Danielewski,On the cancellation problem and automorphism groups of affine algebraic varieties, preprint, Warsaw, 1989.Google Scholar
  5. [DG1] V. Danilov and M. Gizatulin,Automorphisms of affine surfaces I, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya39 (1975), 523–565; English translation: Mathematics of the USSR-Izvestiya9 (1975), 493–534.MathSciNetGoogle Scholar
  6. [DG2] V. Danilov and M. Gizatulin,Automorphisms of affine surfaces II, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya41 (1977), 54–103; English translation: Mathematics of the USSR-Izvestiya11 (1977), 51–98.MATHMathSciNetGoogle Scholar
  7. [DF] J Deveney and D. Finston,Fields of G a-invariants are ruled, Canadian Mathematical Bulletin37 (1994), 37–41.MATHMathSciNetGoogle Scholar
  8. [Di] J. Dixmier,Sur les algebras de Weyl, Bulletin de la Société Mathématique de France96 (1968), 209–242.MATHMathSciNetGoogle Scholar
  9. [F] K.-H. Fieseler,On complex affine surfaces with ℂ +-action. Commentarii Mathematici Helvetici69 (1994), 5–27.MATHCrossRefMathSciNetGoogle Scholar
  10. [FLN] M. Ferrero, Y. Lequain and A. Nowicki,A note on locally nilpotent derivations, Journal of Pure and Applied Algebra79 (1992), 45–50.MATHCrossRefMathSciNetGoogle Scholar
  11. [J] H. W. E. Jung,Uber ganze birationale Transformationen der Eben, Journal für die Reine und Angewandte Mathematik184 (1942), 161–174.MATHCrossRefGoogle Scholar
  12. [KKMLR] S. Kaliman, M. Koras, L. Makar-Limanov and P. Russell,C *-actions on C 3 are linearizable, Electronic Research Announcements of the American Mathematical Society3 (1997), 63–71.MATHCrossRefMathSciNetGoogle Scholar
  13. [KML] S. Kaliman and L. Makar-Limanov,On the Russell-Koras contractible threefolds, Journal of Algebraic Geometry,6 (1997), 247–268.MATHMathSciNetGoogle Scholar
  14. [ML1] L. Makar-Limanov,On groups of automorphisms of a class of surfaces Israel Journal of Mathematics69 (1990), 250–256.MATHMathSciNetCrossRefGoogle Scholar
  15. [ML2] L. Makar-Limanov,On the hypersurface x+x 2y+z2+t3=0 in ℂ4 or a ℂ3 threefold which is not ℂ3, Israel Journal of Mathematics96 (1996), 419–429.MATHMathSciNetGoogle Scholar
  16. [R] R. Rentschler,Operations du groupe additif sur le plane affine, Comptes Rendus de l’Académie des Sciences, Paris267 (1968), 384–387.MATHMathSciNetGoogle Scholar
  17. [Sn] D. Snow,Unipotent actions on affine spaces, inTopological Methods in Algebraic Transformation Groups (H. Kraft et al., eds.), Progress in Mathematics, Vol. 80, Birkhäuser Verlag, Basel-Boston, 1989, pp. 165–176.Google Scholar
  18. [vdK] W. van der Kulk,On polynomial rings in two variables, Nieuw Archief voor Wiskunde1 (1953), 33–41.MATHGoogle Scholar
  19. [W] J. Wilkens,On the cancellation problem for surfaces, Comptes Rendus de l’Académie des Sciences, Paris326 (1998), 1111–1116.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2001

Authors and Affiliations

  • L. Makar-Limanov
    • 1
    • 2
  1. 1.Department of Mathematics and Computer ScienceBar-Ilan UniversityRamat-GanIsrael
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA

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