Skip to main content
SpringerLink
Log in

Finite Subgroups of the Plane Cremona Group

  • Chapter
  • First Online:
Algebra, Arithmetic, and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 269))

Summary

This paper completes the classic and modern results on classification of conjugacy classes of finite subgroups of the group of birational automorphisms of the complex projective plane.

2000 Mathematics Subject Classifications: 14E07 (Primary); 14J26, 14J50, 20B25 (Secondary)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The author was supported in part by NSF grant 0245203.

  2. 2.

    The author was supported in part by RFBR 05-01-00353-a RFBR 08-01-00395-a, grant CRDF RUMI 2692-MO-05 and grant of NSh 1987-2008.1.

  3. 3.

    We thank V. Tsygankov for this observation.

  4. 4.

    We thank J. Blanc for pointing out a mistake in the statement of this theorem in an earlier version of this paper. The correct statement had appeared first in his paper [7].

  5. 5.

    A better argument due to I. Cheltsov shows that in this and the previous cases B is conjugate to a group of automorphisms of \({\mathbb{P}}^2\), or \({\mathbf{F}}_0\), or \({\mathbf{F}}_2\).

References

  1. A. Adem, J. Milgram, Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften, 309, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  2. M. Alberich-CarramiÃČÂćÃćâĂŽÂňÃćâĆňÅŞana, Geometry of the plane Cremona maps, Lecture Notes in Mathematics, 1769. Springer-Verlag, Berlin, 2002.

    Google Scholar 

  3. L. Autonne, Recherches sur les groupes d'ordre fini contenus dans le groupe Cremona, Premier Mémoire Généralités et groupes quadratiques, J. Math. Pures et Appl., (4) 1 (1885), 431–454.

    Google Scholar 

  4. S. Bannai, H. Tokunaga, A note on embeddings of \(S_4\) and \(A_5\) into the Cremona group and versal Galois covers, Publ. Res. Inst. Math. Sci. 43 (2007), 1111–1123.

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Bayle, A. Beauville, Birational involutions of \(P^2\), Asian J. Math. 4 (2000), 11–17.

    MATH  MathSciNet  Google Scholar 

  6. J. Blanc, Finite abelian subgroups of the Cremona group of the plane, thesis, Univ. of Geneva, 2006.

    Google Scholar 

  7. J. Blanc, Elements and cyclic subgroups of finite order of the Cremona group, to appear in Com. Math. Helv.

    Google Scholar 

  8. A. Beauville, J. Blanc, On Cremona transformations of prime order, C. R. Math. Acad. Sci. Paris 339 (2004), 257–259.

    MATH  MathSciNet  Google Scholar 

  9. A. Beauville, p-elementary subgroups of the Cremona group, J. Algebra, 314 (2007), 553–564.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Bertini, Ricerche sulle trasformazioni univoche involutorie nel piano, Annali di Mat. Pura Appl. (2) 8 (1877), 254–287.

    MathSciNet  Google Scholar 

  11. H. Blichfeldt, Finite collineation groups, with an introduction to the theory of operators and substitution groups, Univ. of Chicago Press, Chicago, 1917.

    Google Scholar 

  12. A. Bottari, Sulla razionalità dei piani multipli \(\{x,y,\sqrt[n]{F(x,y)}\}\), Annali di Mat. Pura Appl. (3) 2 (1899), 277–296.

    Google Scholar 

  13. A. Calabri, Sulle razionalità dei piani doppi e tripli cyclici, Ph.D. thesis, Univ. di Roma “La Sapienza”, 1999.

    Google Scholar 

  14. A. Calabri, On rational and ruled double planes, Annali. di Mat. Pura Appl. (4) 181 (2002), 365–387.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Carter, Conjugacy classes in the Weyl group. in “Seminar on Algebraic Groups and Related Finite Groups”, The Institute for Advanced Study, Princeton, N.J., 1968/69, pp. 297–318, Springer, Berlin.

    Google Scholar 

  16. G. Castelnuovo, Sulle razionalità delle involutioni piani, Math. Ann, 44 (1894), 125–155.

    Article  MathSciNet  Google Scholar 

  17. G. Castelnuovo, F. Enriques, Sulle condizioni di razionalità dei piani doppia, Rend. Circ. Mat. di Palermo, 14 (1900), 290–302.

    Article  MATH  Google Scholar 

  18. A. Coble, Algebraic geometry and theta functions (reprint of the 1929 edition), A.M.S. Coll. Publ., v. 10. A.M.S., Providence, RI, 1982. MR0733252 (84m.14001)

    Google Scholar 

  19. J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson, Atlas of finite groups, Oxford Univ. Press, Eynsham, 1985.

    MATH  Google Scholar 

  20. J. Conway, D. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd., Natick, MA, 2003.

    MATH  Google Scholar 

  21. J. Coolidge, A treatise on algebraic plane curves, Dover Publ. New York. 1959.

    MATH  Google Scholar 

  22. A. Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), 223–254.

    MATH  MathSciNet  Google Scholar 

  23. T. de Fernex, On planar Cremona maps of prime order, Nagoya Math. J. 174 (2004), 1–28.

    MATH  MathSciNet  Google Scholar 

  24. T. de Fernex, L. Ein, Resolution of indeterminacy of pairs, in “Algebraic geometry”, pp. 165–177, de Gruyter, Berlin, 2002.

    Google Scholar 

  25. M. Demazure, Surfaces de Del Pezzo, I–V, in “Séminaire sur les Singularités des Surfaces”, ed. by M. Demazure, H. Pinkham and B. Teissier. Lecture Notes in Mathematics, 777. Springer, Berlin, 1980, pp. 21–69.

    Google Scholar 

  26. I. Dolgachev, Weyl groups and Cremona transformations. in “Singularities, Part 1 (Arcata, Calif., 1981)”, 283–294, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983.

    Google Scholar 

  27. I. Dolgachev, Topics in classical algebraic geometry, Part I, manuscript in preparations, see www.math.lsa.umich.edu/idolga/lecturenotes.html.

  28. P. Du Val, On the Kantor group of a set of points in a plane, Proc. London Math. Soc. 42 (1936), 18– 51.

    Article  MATH  Google Scholar 

  29. F. Enriques, Sulle irrazionalita da cui puo farsi dipendere la resoluzione d'un' equazione algebrica \(f(x,y,z) = 0\) con funzioni razionali di due parametri, Math. Ann. 49 (1897), 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  30. C. Geiser, Über zwei geometrische Probleme, J. Reine Angew. Math. 67 (1867), 78–89.

    Article  MATH  Google Scholar 

  31. D. Gorenstein, Finite groups, Chelsea Publ. Co., New York, 1980.

    MATH  Google Scholar 

  32. É. Goursat, Sur les substitutions orthogonales et les divisions rÃČâĂęÃĆ¡guliÃČâĂŽÃĆÂŔres de l'espace, Ann. de e'Ècole Norm. Sup. (3) 6 (1889), 9–102.

    MathSciNet  Google Scholar 

  33. R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.

    Google Scholar 

  34. T. Hosoh, Automorphism groups of cubic surfaces, J. Algebra 192 (1997), 651–677.

    Article  MATH  MathSciNet  Google Scholar 

  35. H. Hudson, Cremona transformations in plane and space, Cambridge Univ. Press. 1927.

    Google Scholar 

  36. V. A. Iskovskih, Rational surfaces with a pencil of rational curves (Russian) Mat. Sb. (N.S.) 74 (1967), 608–638.

    MathSciNet  Google Scholar 

  37. V. A. Iskovskikh, Rational surfaces with a pencil of rational curves with positive square of the canonical class, Math. USSR Sbornik, 12 (1970), 93–117.

    Google Scholar 

  38. V. A. Iskovskih, Minimal models of rational surfaces over arbitrary fields (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 19–43,

    MathSciNet  Google Scholar 

  39. V. A. Iskovskikh, Factorization of birational mappings of rational surfaces from the point of view of Mori theory (Russian) Uspekhi Mat. Nauk 51 (1996), 3–72; translation in Russian Math. Surveys 51 (1996), 585–652.

    MathSciNet  Google Scholar 

  40. V. A. Iskovskikh, Two nonconjugate embeddings of the group \(S_3\times Z_2\) into the Cremona group (Russian) Tr. Mat. Inst. Steklova 241 (2003), Teor. Chisel, Algebra i Algebr. Geom., 105–109; translation in Proc. Steklov Inst. Math. 241 (2003), 93–97.

    MathSciNet  Google Scholar 

  41. V. A. Iskovskikh, Two non-conjugate embeddings of \(S_3\times {\mathbb{Z}}_2\) into the Cremona group II, Adv. Study in Pure Math. 50 (2008), 251–267.

    MathSciNet  Google Scholar 

  42. S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Berlin. Mayer & MÃČâĂęÃĆÂÿller. 111 S. gr. \(8^\circ\). 1895.

    Google Scholar 

  43. J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.

    Google Scholar 

  44. N. Lemire, V. Popov, Z. Reichstein, Cayley groups, J. Amer. Math. Soc. 19 (2006), 921–967.

    Article  MATH  MathSciNet  Google Scholar 

  45. Yu. I. Manin, Rational surfaces over perfect fields. II (Russian) Mat. Sb. (N.S.) 72 (1967), 161–192.

    MathSciNet  Google Scholar 

  46. Yu. I. Manin, Cubic forms: algebra, geometry, arithmetic Translated from Russian by M. Hazewinkel. North-Holland Mathematical Library, Vol. 4. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974.

    Google Scholar 

  47. M. Noether, Über die ein-zweideutigen Ebenentransformationen, Sitzungberichte der physic-medizin, Soc. zu Erlangen, 1878.

    Google Scholar 

  48. B. Segre, The non-singular cubic surface, Oxford Univ. Press. Oxford. 1942.

    Google Scholar 

  49. T. Springer, Invariant theory. Lecture Notes in Mathematics, Vol. 585. Springer-Verlag, Berlin-New York, 1977.

    Google Scholar 

  50. A. Wiman, Zur Theorie endlichen Gruppen von birationalen Transformationen in der Ebene, Math. Ann. 48 (1896), 195–240.

    Article  MATH  MathSciNet  Google Scholar 

  51. H. Zassenhaus, The theory of groups, Chelsea Publ. New York, 1949.

    MATH  Google Scholar 

  52. D.-Q. Zhang, Automorphisms of finite order on rational surfaces. With an appendix by I. Dolgachev, J. Algebra 238 (2001), 560–589.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 525 E. University Ave., Ann Arbor, MI, 49109, USA

    Igor V. Dolgachev & Vasily A. Iskovskikh

Authors
  1. Igor V. Dolgachev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vasily A. Iskovskikh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor V. Dolgachev .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

  2. Eberly College of Science, Dept. Mathematics, Pennsylvania State University, University Park, 16802, Pennsylvania, USA

    Yuri Zarhin

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Busines Meida, LLC

About this chapter

Cite this chapter

Dolgachev, I.V., Iskovskikh, V.A. (2009). Finite Subgroups of the Plane Cremona Group. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_11

Download citation

Publish with us

Policies and ethics

Search

Navigation