European Journal of Mathematics

, Volume 5, Issue 3, pp 1090–1105 | Cite as

G-birational rigidity of the projective plane

  • Dmitrijs SakovicsEmail author
Research Article


Given a surface S and a finite group G of automorphisms of S, consider the birational maps \(S\dashrightarrow S'\) that commute with the action of G. This leads to the notion of a G-minimal variety. A natural question arises: for a fixed group G, is there a birational G-map between two different G-minimal surfaces? If no such map exists, the surface is said to be G-birationally rigid. This paper determines the G-rigidity of the projective plane for every finite subgroup \(G\subset \mathrm {PGL}_{3}(\mathbb {C})\).


Cremona group Birational rigidity Minimal surfaces 

Mathematics Subject Classification

14E07 14J45 20C25 


  1. 1.
    Blichfeldt, H.F.: Finite Collineation Groups. The University of Chicago Press, Chicago (1917)Google Scholar
  2. 2.
    Cheltsov, I., Shramov, C.: Cremona Groups and the Icosahedron. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2016)zbMATHGoogle Scholar
  3. 3.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Eynsham (1985) (With computational assistance from J.G. Thackray)Google Scholar
  4. 4.
    Corti, A.: Factoring birational maps of threefolds after Sarkisov. J. Algebraic Geom. 4(2), 223–254 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dolgachev, I.V., Iskovskikh, V.A.: Finite subgroups of the plane Cremona group. In: Tschinkel, Y., Zarhin, Y. (eds.) Algebra, Arithmetic, and Geometry: : In Honor of Yu. I. Manin. Vol. I. Progress in Mathematics, vol. 269, pp. 443–548. Birkhäuser, Boston (2009)CrossRefGoogle Scholar
  6. 6.
    Hosoh, T.: Automorphism groups of cubic surfaces. J. Algebra 192(2), 651–677 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.8.8 (2017)Google Scholar
  8. 8.
    Yau, S.S.-T., Yu, Y.: Gorenstein Quotient Singularities in Dimension Three. Memoirs of the American Mathematical Society, vol. 105(505). American Mathematical Society, Providence (1993)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Geometry and PhysicsInstitute for Basic Science (IBS)PohangKorea

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