Skip to main content

Advertisement

SpringerLink
Log in

On the Arakelov inequality in positive characteristic

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this note, we generalize the Arakelov inequality in positive characteristic for non-isotrivial semistable families of curves of \(g\ge 2\) which are liftable to \(W_2(k)\) (resp. W(k)). As a consequence, we give an analogue of Beauville’s conjecture in positive characteristic: there are at least 5 singular fibers for non-isotrivial semistable families of curves of \(g\ge 2\) over \(\mathbb {P}^1\) which are liftable to W(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. Arakelov, S.J.: Families of algebraic curves with fixed degeneracy. Math. USSR Izv. 5, 1277–1302 (1971)

    Article  MATH  Google Scholar 

  2. Beauville, A.: Le nombre minimum de fibres singulières d’un courbe stable sur \({\mathbb{P}}^1\), In: Séminaire sur les Pinceaux de Courbes de Genre au Moins Deux. Astérisque, Société Mathématique de France, Paris, 86, 97–108 (1981)

  3. Cornalba, M., Harris, J.: Divisor classes associated to families of stable varieties, with application to the moduli space of curves. Ann. Sci. Ecol. Norm. Sup. 21, 455–475 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deligne, P., Illusie, L.: Relèvements modulo \(p^2\) et decomposition du complexe de de Rham. Invent. Math. 89, 247–270 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems. DMV-Seminar, vol. 20. Birkhäuser, Basel (1992)

    Book  MATH  Google Scholar 

  6. Faltings, G.: Arakelov’s theorem for abelian varieties. Invent. Math. 73, 337–347 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kukulies, S.: On Shimura curves in the Schottky locus. J. Algebra. Geom. 19, 371–397 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kutsura, T., Ueno, K.: On elliptic surfaces in characteristic \(p\). Math. Ann. 272, 291–330 (1985)

    Article  MathSciNet  Google Scholar 

  9. Langer, A.: Bogomolov’s inequality for Higgs sheaves in positive characteristic. Invent. Math. 199, 889–920 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Langer, A.: The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic. Duke Math. J. 165, 2737–2769 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liedtke, C.: Algebraic surfaces in positive characteristic. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds.) Birational geometry, rational curves, and arithmetic, pp. 229–292. Springer, New York (2013)

  12. Liu, K.-F.: Geometric height inequalities. Math. Res. Lett. 3(5), 693–702 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lu, J., Sheng, M., Zuo, K.: An Arakelov inequality in characteristic \(p\) and upper bound of \(p\)-rank zero locus. J. Number Theory 129(12), 3029–3045 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lu, X., Tan, S.-L., Xu, W.-Y., Zuo, K.: On the minimal number of singular fibers with non-compact Jacobians for families of curves over \(\mathbb{P}^1\). J. Math. Pure. Appl. 105(5), 724–733 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lu, J., Tan, S.-L., Yu, F., Zuo, K.: A new inequality on the Hodge number \(h^{1,1}\) of algebraic surfaces. Math. Zeit. 276, 543–555 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lu, X., Zuo, K.: The Oort conjecture on Shimura curves in the Torelli locus of curves. arXiv:1405.4751v2

  17. Lu, X., Zuo, K.: The Oort conjecture on Shimura curves in the Torelli locus of hyperelliptic curves. J. Math. Pure. Appl. 108(4), 532–552 (2017)

  18. Moret-Bailly, L.: Familles de courbes et de variétés abéliennes sur \(\mathbb{P}^1\). Astérisque 86, 125–140 (1981)

    MATH  Google Scholar 

  19. Moriwaki, A.: Bogomolov conjecture over function fields for stable curves with only irreducible fibers. Compos. Math. 105(2), 125–140 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Möller, M.: Shimura and Teichmüller curves. J. Mod. Dyn. 5(1), 1–32 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nguyen, K.V.: A remark on semi-stable fibrations over \(\mathbb{P}^1 \) in positive characteristic. Compos. Math. 112(1), 41–44 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127, 309–316 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Serre, J.-P.: Sur la topologie des variétés algébriques en charactéristique \(p\). In: Mexico, Symposion Internacional de Topologia Algebraica, pp. 24–53 (1958)

  24. Sheng, M., He, X., Zuo, K.: A note on the characteristic \(p\) nonabelian Hodge theory in the geometric case. Int. J. Math. 26(1), 18 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tan, S.-L.: The minimal number of singular fibers of a semistable curve over \(\mathbb{P}^1\). J. Algebra Geom. 4(3), 591–596 (1995)

    MathSciNet  MATH  Google Scholar 

  26. Tan, S.-L., Tu, Y.-P., Zamora, A.-G.: On complex surfaces with 5 or 6 semistable singular fibers over \(\mathbb{P}^1\). Math. Zeit. 249, 427–438 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tan, S.-L., Xu, W.-Y.: On Szpiro inequality for semistable families of curves. J. Number Theory 151, 36–45 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Viehweg, E., Zuo, K.: On the isotriviality of families of projective manifolds over curves. J. Algebra Geom. 10, 781–799 (2001)

    MathSciNet  MATH  Google Scholar 

  29. Viehweg, E., Zuo, K.: A characterization of certain Shimura curves in the moduli stack of abelian varieties. J. Differ. Geom. 66(2), 233–287 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Viehweg, E., Zuo, K.: Numerical bounds for semi-stable families of curves or of certain higher dimensional manifolds. J. Algebra Geom. 15, 771–791 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Viehweg, E., Zuo, K.: Arakelov inequalities and the uniformization of certain rigid Shimura varieties. J. Differ. Geom. 77(2), 291–352 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann. 276, 449–466 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yuan, X., Zhang, T.: Relative Noether inequality on fibered surfaces. Adv. Math. 259, 89–115 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Prof. Adrian Langer, Christian Liedtke, Sheng-Li Tan, Kang Zuo and Dr. Xin Lu, Tong Zhang for useful conversations.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Wan-Yuan Xu

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Wan-Yuan Xu

Authors
  1. Wan-Yuan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wan-Yuan Xu.

Additional information

This work is supported by NSFC (No. 11601088) and China Postdoctoral Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, WY. On the Arakelov inequality in positive characteristic. Math. Z. 289, 109–117 (2018). https://doi.org/10.1007/s00209-017-1945-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-017-1945-5

Keywords

Mathematics Subject Classification

Search

Navigation