• Yasinsky, Egor (Steklov Mathematical Institute of Russian Academy of Sciences)
  • Received : 2016.10.07
  • Accepted : 2017.02.13
  • Published : 2017.09.30


We compute the Jordan constant for the group of birational automorphisms of a projective plane ${\mathbb{P}}^2_{\mathbb{k}}$, where ${\mathbb{k}}$ is either an algebraically closed field of characteristic 0, or the field of real numbers, or the field of rational numbers.


  1. C. Birkar, Singularities of linear systems and boundedness of Fano varieties,
  2. M. J. Collins, On Jordans theorem for complex linear groups, J. Group Theory 10 (2007), no. 4, 411-423.
  3. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative al-gebras, Pure and Applied Mathematics, Vol. XI. Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962.
  4. I. V. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, 1st edition, 2012.
  5. I. V. Dolgachev and V. A. Iskovskikh, Finite subgroups of the plane Cremona group, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Progr. Math., vol. 269 (2009), Birkhauser Boston, Inc., Boston, MA., 443-558.
  6. I. V. Dolgachev and V. A. Iskovskikh, On elements of prime order in the plane Cremona group over a perfect field, Int. Math. Res. Notices 2009 (2009), no. 18, 3467-3485.
  7. T. Hosoh, Automorphism groups of quartic del Pezzo surfaces, J. Algebra 185 (1996), no. 2, 374-389.
  8. T. Hosoh, Automorphism groups of cubic surfaces, J. Algebra 192 (1997), no. 2, 651-677.
  9. I. M. Isaacs, Finite group theory, volume 92 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2008.
  10. J. Kollar, Real Algebraic Surfaces, Notes of the 1997 Trento summer school lectures, preprint.
  11. J. Patera and Y. Saint-Aubin, Finite subgroups of the Lorentz group and their generating functions, Symmetries in Science, Springer (1980), 297-308.
  12. V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, Affine algebraic geometry, 289-11, CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence, RI, 2011.
  13. V. L. Popov, Finite subgroups of diffeomorphism groups, Proc. Steklov Inst. Math. 289 (2015), no. 1, 221-226.
  14. Yu. Prokhorov and C. Shramov, Jordan property for Cremona groups, Amer. J. Math. 138 (2016), no. 2, 403-418.
  15. Yu. Prokhorov and C. Shramov, Jordan constant for Cremona group of rank 3,
  16. M. F. Robayo, Prime order birational diffeomorphisms of the sphere, Annali Sc. Norm. Super. Pisa, Cl. Sci. (5) XVI (2016), 909-970.
  17. J.-P. Serre, Le groupe de Cremona et ses sous-groupes finis, Seminaire Bourbaki 1000 (2008), 75-100.
  18. J.-P. Serre, A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field, Mosc. Math. J. 9 (2009), no. 1, 193-208.
  19. A. Trepalin, Rationality of the quotient of ${\mathbb{P}}^2$ by finite group of automorphisms over arbitrary field of characteristic zero, Cent. Eur. J. Math. 12 (2014), no. 2, 229-239.
  20. E. Yasinsky, Subgroups of odd order in the real plane Cremona group, J. Algebra 461 (2016), 87-120.
  21. Yu. Prokhorov and C. Shramov, Finite groups of birational selfmaps of threefolds,
  22. Yu. Prokhorov and C. Shramov, p-subgroups in the space Cremona group,