Skip to main content
SpringerLink
Log in

Seshadri constants on ruled surfaces: the rational and the elliptic cases

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We study the Seshadri constants on geometrically ruled surfaces. The unstable case is completely solved. Moreover, we give some bounds for the stable case. We apply these results to compute the Seshadri constant of the rational and elliptic ruled surfaces. Both cases are completely determined. The elliptic case provides an interesting picture of how particular is the behavior of the Seshadri constants.

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. Atiyah, M.F.: Vector bundles over an elliptic curve. Proc. London Math. Soc. (3) VII 27, 414–452 (1957)

  2. Bauer, T.: Seshadri constants and periods of polarized abelian varieties. Math. Ann. 312, 607–623 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bauer, T.: Seshadri constants on algebraic surfaces. Math. Ann. 313, 547–583 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Demailly, J.-P.: Singular hermitian metrics on positive line bundles Lecture Notes Math. 1507, 87–104 (1992)

  5. Fuentes García, L., Pedreira, M.: The projective theory of ruled surfaces. Note Mat., 24, 25–63 (2005)

    MATH  MathSciNet  Google Scholar 

  6. Fuentes García, L.: Classification of elliptic lines scrolls. Preprint arXiv:math.AG/0009050

  7. Ein, L; Küchle, O., Lazarsfeld, R.: Local positivity of ample line bundles. J. Differential Geom. 42, N.2, 193–219 (1995)

    Google Scholar 

  8. Hartshorne, R.: Algebraic Geometry. GTM, 52. Springer–Verlag, 1977

  9. Lanteri, A., Turrini, C.: On the automorphism of elliptic ruled surfaces. Boll. Un. Mat. Ital. D (6) I, N.1, 41–58 (1982)

    Google Scholar 

  10. Nakamaye, M.: Seshadri constants at very general points. Trans. Amer. Math. Soc. 357, N.8, 3285–3297 (2005)

    Google Scholar 

  11. Schultz, Ch.: Seshadri constants on abelian surfaces. Thesis, Marburg, 2004.

  12. Steffens, A.: Remarks on Seshadri constants. Math. Z. 227, 505–510 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Xu, G.: Ample line bundles on smooth surfaces. J. reine angew. Math. 469, 199–209 (1995)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Métodos Matemáticos y Representación. E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad de A Coruña, Campus de Elviña, 15192 A, Coruña, Spain

    Luis Fuentes García

Authors
  1. Luis Fuentes García
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis Fuentes García.

Rights and permissions

Reprints and permissions

About this article

Cite this article

García, L. Seshadri constants on ruled surfaces: the rational and the elliptic cases. manuscripta math. 119, 483–505 (2006). https://doi.org/10.1007/s00229-006-0629-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-006-0629-y

Keywords

Search

Navigation