Skip to main content
SpringerLink
Log in

On \(W_2\)-lifting of Frobenius of algebraic surfaces

  • Published:
Collectanea Mathematica Aims and scope Submit manuscript

Abstract

The classification of minimal algebraic surfaces in positive characteristics was accomplished by Bombieri and Mumford in the 1970s. In this work we decide completely which minimal algebraic surfaces in positive characteristics allow a lifting of their Frobenius over the truncated Witt rings of length 2. Besides, we show that the Frobenius morphism of many projective rational surfaces cannot be lifted to \(W_2(k)\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beauville, A.: Complex algebraic surfaces, vol. 34. Cambridge University Press (1996)

  2. Beauville, A.: Endomorphisms of hypersurfaces and other manifolds. Int. Math. Res. Notices 2001(1), 53–58 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Blass, P.: Unirationality of enriques surfaces in characteristic two. Compos. Math. 45(3), 393–398 (1982)

    MATH  MathSciNet  Google Scholar 

  4. Bogomolov, F., Hassett, B., Tschinkel, Y.: Constructing rational curves on k3 surfaces. Duke Math. J. 157(3), 535–550 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char. \(p\), iii. Invent. Math. 35(1), 197–232 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char. p ii. Complex analysis and algebraic geometry, Collected Papers dediccated to K. Kodaira, Iwanami Shoten, Tokyo. pp. 23–42 (1977)

  7. Buch, A., Thomsen, J.F., Lauritzen, N., Mehta, V.: The frobenius morphism on a toric variety. Tohoku Math. J. 49, 355–366 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cossec, F.R., Dolgachev, I.: Enriques surfaces, vol. 1. Birkhäuser Boston (1989)

  9. Crew, R.: Etale \(p\)-covers in characteristic \(p\). Compos. Math. 52(1), 31–45 (1984)

    MATH  MathSciNet  Google Scholar 

  10. Cynk, S., van Straten, D.: Small resolutions and non-liftable calabi-yau threefolds. Manuscr. Math. 130(2), 233–249 (2009)

    Article  MATH  Google Scholar 

  11. Deligne, P., Illusie, L.: Relèvements modulop 2 et décomposition du complexe de de rham. Invent. Math. 43(2), 247–270 (1987)

    Article  MathSciNet  Google Scholar 

  12. Dupuy, T.: Positivity and lifts of the frobenius. preprint (2012)

  13. Hahn, D.W., Miranda, R.: Quadruple covers of algebraic varieties. Ph.D. thesis, Colorado State University (1996)

  14. Hartshorne, R.: Algebraic geometry, vol. 52. Springer (1977)

  15. Illusie, L.: Frobenius and hodge degeneration. Introd. Hodge Theory 8, 99–149 (2002)

    MathSciNet  Google Scholar 

  16. Illusie, L.: Grothendieck’s existence theorem in formal geometry with a letter of jean-pierre serre. Math. Surv. Monogr. 123, 179 (2005)

    MathSciNet  Google Scholar 

  17. Kunz, E.: Kähler differentials, advanced lectures in mathematics, friedr. Vieweg, Braunschweig (1986)

    Book  Google Scholar 

  18. Lang, W.E.: Quasi-elliptic surfaces in characteristic three. Ann. Sci. de l’École Normale Supérieure 12(4), 473–500 (1979)

    MATH  Google Scholar 

  19. Lange, H., Stuhler, U.: Vektorbündel auf kurven und darstellungen der algebraischen fundamentalgruppe. Math. Z. 156(1), 73–83 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  20. Liedtke, C.: Algebraic surfaces in positive characteristic. Birational geometry, rational curves, and arithmetic. Springer, New York. pp. 229–292 (2013)

  21. Liedtke, C., Satriano, M.: On the birational nature of lifting. Adv. Math. 254, 118–137 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mehta, V., Srinivas, V.: Varieties in positive characteristic with trivial tangent bundle. Compos. Math. 64(2), 191–212 (1987)

    MATH  MathSciNet  Google Scholar 

  23. Miranda, R.: Triple covers in algebraic geometry. Am. J. Math. 107(5), 1123–1158 (1985)

    Article  MATH  Google Scholar 

  24. Oda, T.: Vector bundles on an elliptic curve. Nagoya Math. J. 43, 41–72 (1971)

    MATH  MathSciNet  Google Scholar 

  25. Paranjape, K.H., Srinivas, V.: Self maps of homogeneous spaces. Invent. Math. 98(2), 425–444 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  26. Raynaud, M.: Around the mordell conjecture for function fields and a conjecture of serge lang in algebraic geometry. In: Proceedings of the Japan–France Conference Held at Tokyo and Kyoto, oct 5–14, 1982 pp. 1–19 (1983)

Download references

Acknowledgments

The author would like to thank Prof. Xiaotao Sun for a careful reading of a draft of this work. The author would also like to thank the reviewers for pointing out mistakes in the original version of this work and also their helpful advice on improving this work.

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Bohai University, No. 19, Keji Rd., New Songshan District, Jinzhou, 121013, LiaoNing, People’s Republic of China

    He Xin

Authors
  1. He Xin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to He Xin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xin, H. On \(W_2\)-lifting of Frobenius of algebraic surfaces. Collect. Math. 67, 69–83 (2016). https://doi.org/10.1007/s13348-014-0130-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13348-014-0130-y

Keywords

Mathematics Subject Classification

Search

Navigation