Advertisement

Rendiconti del Circolo Matematico di Palermo

, Volume 62, Issue 1, pp 111–125 | Cite as

Relations between some invariants of algebraic varieties in positive characteristic

  • Gerard van der  GeerEmail author
  • Toshiyuki Katsura
Article
  • 117 Downloads

Abstract

We discuss relations between certain invariants of varieties in positive characteristic, like the \(a\)-number and the height of the Artin-Mazur formal group. We calculate the \(a\)-number for Fermat surfaces.

Keywords

a-number b-number Height Hodge filtration Conjugate filtration  Fermat surface 

Mathematics Subject Classification (1991)

14G 14G17 11G25 

Notes

Acknowledgments

The second author was supported in part by JSPS Grant-in-Aid (S), No 19104001.

References

  1. 1.
    Artin, M.: Supersingular K3 surfaces. Ann. Sci. ENS $4^{e}$ série T. 4, fasc 4, 543–568 (1974)Google Scholar
  2. 2.
    Artin, M., Mazur, B.: Formal groups arising from algebraic varieties. Ann. Sci. ENS $4^{e}$ série T. 10, fasc. 4, 87–132 (1977)Google Scholar
  3. 3.
    Chatzistamatiou, A.: Commutative formal groups arising from schemes. arXiv:1203.4926 (2012)Google Scholar
  4. 4.
    Deligne, P., Illusie, L.: Relévements modulo $p^2$ et décomposition du complexe de de Rham. Invent. Math. 89, 247–270 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    van der Geer, G., Katsura, T.: On a stratification of the moduli of K3 surfaces. J. Eur. Math. Soc. 2, 259–290 (2000)zbMATHCrossRefGoogle Scholar
  6. 6.
    van der Geer, G., Katsura, T.: Formal Brauer groups and moduli of abelian surfaces. In: Moduli of Abelian varieties, Progress in Math. vol. 195, pp. 185–202, Springer, Birkhauser (2001)Google Scholar
  7. 7.
    van der Geer, G., Katsura, T.: An invariant for algebraic varieties in positive characteristic. Contemp. Math. 300, 131–141 (2002)CrossRefGoogle Scholar
  8. 8.
    van der Geer, G., Katsura, T.: On the height of Calabi-Yau varieties in positive characteristic. Documenta Mathematica 8, 97–113 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ibukiyama, T., Katsura, T., Oort, F.: Supersingular curves of genus two and class numbers. Comp. Math. 57, 127–152 (1986)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Illusie, L.: Complexe de De Rham-Witt. Astérisque 63, 83–112 (1979)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Illusie, L.: Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. ENS 12(4), 501661 (1979)MathSciNetGoogle Scholar
  12. 12.
    Katz, N.: Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration). Invent. Math. 18, 1–118 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Koblitz, N.: p-adic variation of the zeta-function over families defined over finite fields. Comp. Math. 31, 119–218 (1975)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Moonen, B., Wedhorn, T.: Discrete invariants of varieties in positive characteristic. Int. Math. Res. Not. 72, 3855–3903 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mumford, D.: Lectures on curves on an algebraic surface. Annals of Mathematics Studies 59. University Press, Princeton (1966)Google Scholar
  16. 16.
    Ogus, A.: Supersingular K3 crystals. Astérisque 64, 3–86 (1979)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Oort, F.: Commutative group schemes. Lecture Notes in Mathematics, vol. 15, Springer, Verlag (1966)Google Scholar
  18. 18.
    Sasakura, N.: On some results on the Picard numbers of certain algebraic surfaces. J. Math. Soc. Jpn. 20, 297–321 (1968)Google Scholar
  19. 19.
    Serre, J.-P.: Sur la topologie des variétés algébriques en caractéristique $p$. In: Symposion Internacional de topologia algebraica, pp. 24–53 (1958)Google Scholar
  20. 20.
    Shioda, T., Katsura, K.: On Fermat varieties. Tohoku Math. J. 31, 97–115 (1979)Google Scholar

Copyright information

© Springer-Verlag Italia 2013

Authors and Affiliations

  1. 1.Korteweg-de Vries InstituutUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Faculty of Science and EngineeringHosei UniversityTokyoJapan

Personalised recommendations