A Stratification on the Moduli of K3 Surfaces in Positive Characteristic

  • Gerard van der GeerEmail author
Part of the Progress in Mathematics book series (PM, volume 319)


We review the results on the cycle classes of the strata defined by the height and the Artin invariant on the moduli of K3 surfaces in positive characteristic obtained in joint work with Katsura and Ekedahl. In addition we prove a new irreducibility result for these strata.


Modulus Space Weyl Group Cycle Class Hodge Class Supersingular Elliptic Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [A]
    M. Artin, Supersingular K3 surfaces. Ann. Sci. École Norm. Sup. 7, 543–567 (1974)MathSciNetzbMATHGoogle Scholar
  2. [A-M]
    M. Artin, B. Mazur, Formal groups arising from algebraic varieties. Ann. Sci. Ecole Norm. Sup. 10, 87–132 (1977)MathSciNetzbMATHGoogle Scholar
  3. [A-SD]
    M. Artin, H.P.F. Swinnerton-Dyer, The Shafarevich-Tate conjecture for pencils of elliptic curves on K3surfaces. Invent. Math. 20, 249–266 (1973)Google Scholar
  4. [C]
    F. Charles, The Tate conjecture for K3 surfaces over finite fields. Invent. Math. 194 (1), 119–145 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [C-O]
    C-L. Chai, F. Oort, Monodromy and irreducibility of leaves. Ann. Math. (2) 173, 1359–1396 (2011)Google Scholar
  6. [E]
    T. Ekedahl, On supersingular curves and abelian varieties. Math. Scand. 60, 151–178 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [E-vdG1]
    T. Ekedahl, G. van der Geer, Cycle classes of the E-O stratification on the moduli of abelian varieties, in Algebra, Arithmetic and Geometry. Progress in Mathematics, vol. 269–270 (Birkhäuser, Boston, 2010)Google Scholar
  8. [E-vdG2]
    T. Ekedahl, G. van der Geer, Cycle Classes on the Moduli of K3 surfaces in positive characteristic. Sel. Math. (N.S.) (2011, to appear). arXiv:1104.3024.
  9. [G-H-S]
    V. Gritsenko, K. Hulek, G. Sankaran, The Hirzebruch-Mumford volume for the orthognal group and applications. Doc. Math. 12, 215–241 (2007)MathSciNetzbMATHGoogle Scholar
  10. [H]
    D. Huybrechts, Lectures on K3 surfaces. Lecture Notes.
  11. [K-S]
    S. Kondo, I. Shimada, On certain duality of Néron-Severi lattices of supersingular K3 surfaces and its application to generic supersingular K3 surfaces. Algebraic Geom. 1 (3), 311–333 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [L]
    C. Liedtke, Supersingular k3 surfaces are unirational. Eprint. arXiv:1304.5623Google Scholar
  13. [M]
    D. Maulik, Supersingular K3 surfaces for large primes. Eprint. arXiv:1203.2889Google Scholar
  14. [Mo]
    B. Moonen, A dimension formula for Ekedahl-Oort strata. Ann. Inst. Fourier (Grenoble) 54 (3), 666–698 (2004)Google Scholar
  15. [M-W]
    B. Moonen, T. Wedhorn, Discrete invariants of varieties in positive characteristic. Int. Math. Res. Not. 2004 (72), 3855–3903 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Og1]
    A. Ogus, Supersingular K3 crystals. Journées de Géométrie Algébrique de Rennes (II). Astérisque, vol. 64 (Société Mathématique de France, Paris, 1979), pp. 3–86.Google Scholar
  17. [Og2]
    A. Ogus, Singularities of the height strata in the moduli of K3 surfaces, Moduli of Abelian Varieties. (Texel Island, 1999). Progress in Mathematics, vol. 195 (Birkhäuser, Basel, 2001), pp. 325–343Google Scholar
  18. [O]
    F. Oort, A stratification of a moduli space of abelian varieties, Moduli of Abelian Varieties (Texel Island, 1999). Progress in Mathematics, vol. 195, (Birkhäuser, Basel, 2001) pp. 345–416Google Scholar
  19. [P1]
    K. Madapusi Pera, Toroidal compactifications of integral models of Shimura varieties of Hodge type. Eprint. arXiv:1211.1731Google Scholar
  20. [P2]
    K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic. Eprint. arXiv:1301.6326Google Scholar
  21. [P-R]
    H. Pittie, A. Ram, A Pieri-Chevalley formula in the K-theory of a GB-bundle. Electron. Res. Announc. Am. Math. Soc. 5, 102–107 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [R]
    M. Rapoport, On the Newton stratification. Séminaire Bourbaki. Astérisque No. 290, vol. 2001/2002, Exp. No. 903, viii, 207–224 (2003)Google Scholar
  23. [R-S2]
    A.N. Rudakov, I.R. Shafarevich, Supersingular K3 surfaces over fields of characteristic 2. Izv. Akad. Nauk SSSR Ser. Mat. 42, 848–869 (1978). Also in: I.R. Shafarevich: Collected Mathematical Papers. Springer Verlag.Google Scholar
  24. [R-S]
    A.N. Rudakov, I.R. Shafarevich, Surfaces of type K3 over fields of finite characteristic. Current Problems in Mathematics, vol. 18 (Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981), pp. 115–207. Also in: I.R. Shafarevich: Collected Mathematical Papers. Springer Verlag.Google Scholar
  25. [R-S-Z]
    A.N. Rudakov, I.R. Shafarevich, Th. Zink, The effect of height on degenerations of K3 surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 46 (1), 117–134, 192 (1982). Also in: I.R. Shafarevich: Collected Mathematical Papers. Springer Verlag.Google Scholar
  26. [S]
    T. Shioda, Algebraic cycles on certain K3 surfaces in characteristic p. Manifolds–Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) (University of Tokyo Press, Tokyo, 1975), pp. 357–364Google Scholar
  27. [T]
    J. Tate, Algebraic cycles and poles of zeta functions. Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) (Harper & Row, New York, 1965) pp. 93–110Google Scholar
  28. [vdG]
    G. van der Geer, Cycles on the moduli space of abelian varieties. Moduli of Curves and Abelian Varieties. Aspects of Mathematics (vol. E33, pp. 65–89) (Vieweg, Braunschweig, 1999)Google Scholar
  29. [vdG-K1]
    G. van der Geer, T. Katsura, On a stratification of the moduli of K3 surfaces. J. Eur. Math. Soc. (JEMS) 2 (3), 259–290 (2000)Google Scholar
  30. [vdG-K2]
    G. van der Geer, T. Katsura, Note on tautological classes of moduli of K3 surfaces. Mosc. Math. J. 5, 775–779 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Korteweg-de Vries InstituutUniversiteit van AmsterdamAmsterdamThe Netherlands

Personalised recommendations