Mathematische Annalen

, Volume 350, Issue 4, pp 757–791 | Cite as

Classification of K3-surfaces with involution and maximal symplectic symmetry

  • Kristina FrantzenEmail author


K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo surfaces, and a complete classification of K3-surfaces with maximal symplectic symmetry is obtained.

Mathematics Subject Classification (2000)

14J28 14J50 14L30 


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  1. 1.
    Bayle L., Beauville A.: Birational involutions of P2. Asian J. Math. 4(1), 11–17 (2000) Kodaira’s issuezbMATHMathSciNetGoogle Scholar
  2. 2.
    Beauville A.: p-Elementary subgroups of the Cremona group. J. Algebra 314(2), 553–564 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beauville A., Blanc J.: On Cremona transformations of prime order. C. R. Math. Acad. Sci. Paris 339(4), 257–259 (2004)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Blanc, J.: Finite abelian subgroups of the Cremona group of the plane. Ph.D. thesis, Université de Genève (2006)Google Scholar
  5. 5.
    Blanc J.: Finite abelian subgroups of the Cremona group of the plane. C. R. Math. Acad. Sci. Paris 344(1), 21–26 (2007)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Blichfeldt H.F.: Finite Collineation Groups. The University of Chicago Press, Chicago (1917)Google Scholar
  7. 7.
    Crass S.: Solving the sextic by iteration: a study in complex geometry and dynamics. Exp. Math. 8(3), 209–240 (1999)zbMATHMathSciNetGoogle Scholar
  8. 8.
    de Fernex T.: On planar Cremona maps of prime order. Nagoya Math. J. 174, 1–28 (2004)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Dolgachev, I.V.: Weighted projective varieties. In: Group Actions and Vector Fields (Vancouver, B.C., 1981). Lecture Notes in Mathematics, vol. 956, pp. 34–71. Springer, Berlin (1982)Google Scholar
  10. 10.
    Dolgachev, I.V.: Topics in Classical Algebraic Geometry. Version August 1, 2010.
  11. 11.
    Dolgachev, I.V., Iskovskikh, V.A.: Finite subgroups of the plane Cremona group. In: Algebra, Arithmetic, and Geometry, vol. I, in honour of Yu. I. Manin. Progress in Mathematics, vol. 269, pp. 443–548. Birkhäuser, Boston (2009)Google Scholar
  12. 12.
    Dolgachev I.V., Iskovskikh V.A.: On elements of prime order in the plane Cremona group over a perfect field. Int. Math. Res. Not. IMRN 18, 3467–3485 (2009)MathSciNetGoogle Scholar
  13. 13.
    Frantzen, K.: K3-surfaces with special symmetry. Ph.D. thesis, Ruhr-Universität Bochum (2008)Google Scholar
  14. 14.
    Frantzen, K., Huckleberry, A.: K3-surfaces with special symmetry: an example of classification by Mori-reduction. In: Complex Geometry in Osaka, in honour of Professor Akira Fujiki on the occasion of his 60th birthday. Lecture Note Series in Mathematics, vol. 9, pp. 86–99. Osaka Mathematical Publications, Osaka (2008)Google Scholar
  15. 15.
    Frantzen K., Huckleberry A.: Finite symmetry groups in complex geometry. Journées Élie Cartan 2006, 2007 et 2008 Nancy, Revue de l’Institute Élie Cartan, Nancy 19, 73–113 (2009)Google Scholar
  16. 16.
    Iskovskikh V.A.: Minimal models of rational surfaces over arbitrary fields. Math. USSR-Izv. 14(1), 17–39 (1980)zbMATHCrossRefGoogle Scholar
  17. 17.
    Keum J.H., Oguiso K., Zhang D.-Q.: The alternating group of degree 6 in the geometry of the Leech lattice and K3 surfaces. Proc. Lond. Math. Soc. (3) 90(2), 371–394 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Keum J.H., Oguiso K., Zhang D.-Q.: Extensions of the alternating group of degree 6 in the geometry of K3 surfaces. Eur. J. Comb. 28(2), 549–558 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kollár, J., Mori, S.: Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)Google Scholar
  20. 20.
    Kondō S.: Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K3 surfaces with an appendix by S. Mukai. Duke Math. J. 92(3), 593–603 (1998)zbMATHCrossRefGoogle Scholar
  21. 21.
    Manin Y.I.: Rational surfaces over perfect fields. II. Math. USSR-Sb. 1(2), 141–168 (1967)zbMATHCrossRefGoogle Scholar
  22. 22.
    Miller G.A., Blichfeldt H.F., Dickson L.E.: Theory and Applications of Finite Groups. Dover, New York (1916)zbMATHGoogle Scholar
  23. 23.
    Mori S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. (2) 116(1), 133–176 (1982)CrossRefGoogle Scholar
  24. 24.
    Mukai S.: Finite groups of automorphisms of K3 surfaces and the Mathieu group. Invent. Math. 94(1), 183–221 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Nikulin V.V.: Kummer surfaces. Math. USSR. Izv. 9(2), 261–275 (1976)zbMATHCrossRefGoogle Scholar
  26. 26.
    Nikulin V.V.: Finite automorphism groups of Kähler K3 surfaces. Trans. Moscow Math. Soc. 38(2), 71–135 (1980)Google Scholar
  27. 27.
    Nikulin V.V.: On factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebrogeometric applications. J. Sov. Math. 22, 1401–1476 (1983)zbMATHCrossRefGoogle Scholar
  28. 28.
    Oguiso, K., Zhang, D.-Q.: The simple group of order 168 and K3 surfaces. In: Complex Geometry (Göttingen, 2000), pp. 165–184. Springer, Berlin (2002)Google Scholar
  29. 29.
    Yau S.S.-T., Yu Y.: Gorenstein quotient singularities in dimension three. Mem. Am. Math. Soc. 105(505), 1–88 (1993)MathSciNetGoogle Scholar
  30. 30.
    Zhang D.-Q.: Quotients of K3 surfaces modulo involutions. Jpn. J. Math. (N.S.) 24(2), 335–366 (1998)zbMATHGoogle Scholar
  31. 31.
    Zhang D.-Q.: Automorphisms of finite order on rational surfaces with an appendix by I. Dolgachev. J. Algebra 238(2), 560–589 (2001)zbMATHCrossRefGoogle Scholar
  32. 32.
    Zhang, D.-Q.: Automorphisms of K3 surfaces. In: Proceedings of the International Conference on Complex Geometry and Related Fields (Providence, RI), AMS/IP Stud. Adv. Math., vol. 39, pp. 379–392. Amer. Math. Soc. (2007)Google Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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