Skip to main content
SpringerLink
Log in

Seshadri constants and Fano manifolds

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

In memory of Meeyoung Kim In this paper, we give a lower bound of Seshadri constants on smooth Fano varieties. More precisely, we show that on a smooth Fano manifold of dimension n whose anticanonical system is base point free, Seshadri constants of ample divisors are bounded from below by one over n−2. As a corollary we recover the earlier result on Fano threefolds.

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. Andreatta, M., Wisniewski, J.: A view on contraction of higher dimensional varieties. Proc. Sympo. Pure. Math. Part1. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 153–184

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Beltrametti, M., Sommese, A.: The adjunction theory of complex projective varieties. de Gruyter Expositions in Mathematics 16, (1995)

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

    Google Scholar 

  5. Ein, L., Lazarsfeld, R.: Seshadri constants on smooth surfaces. Astérisque 218, 177–186 (1993)

    MATH  Google Scholar 

  6. Ein, L., Lazarsfeld, R.: Global generation of pluricanonical and adjoint linear series on smooth projective threefold. J. Amer. Math. Soc. 6, 875–903 (1993)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Fujita, T.: Birational geometry of algebraic varieties: Open problem. The 23rd International Symposium of the Division of Mathematics of the Taniguchi Foundation, Katata, 1988, pp. 42–45

  9. Fujita, T.: Classification theories of polarized varieties. LMS Lecture Note Series Vol. 155, Cambridge University Press, 1990

  10. Fujita, T.: Remarks on Ein-Lazarsfeld criterion of spannedness of adjoint bundles of polarized threefold. Alg-geom eprint 9311013

  11. Helmke, S.: ICTP lecture note (2000)

  12. Hwang, J.-M., Keum, J.: Seshadri-exceptional foliations. Preprint

  13. Iskovskikh, V.A.: Fano 3-folds I. Math. USSR-Izv 11, 485–527 (1977)

    MATH  Google Scholar 

  14. Iskovskikh, V.A., Shokurov, V.V.: Biregular theory of Fano 3-folds. LNM 732, 171–182 (1979)

    MATH  Google Scholar 

  15. Kawamata, Y.: On Fujita’s freeness conjecture for 3-folds and 4-folds. Math. Ann. 308, 491–505 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. Adv. Stud. Pure. Math. 10, North-Holland, Amsterdam-New York, 1987, pp. 283–360

  17. Kollár, J.: Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 32. Springer-Verlag, Berlin

  18. Lee, S.: Linear system on Fano threefold. Preprint

  19. Mori, S., Mukai, S.: On Fano threefolds with B 2(X)≥2. Adv. Stud. Pure Math. 1, 101–130 (1982)

    MATH  Google Scholar 

  20. Nakamaye, M.: Seshadri constants on abelian varieties. Amer. J. Math. 118, 621–635 (1996)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Konkuk University (143-701), Kwangjin-Gu Hwayang-dong 1, Seoul, Korea

    Seunghun Lee

Authors
  1. Seunghun Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seunghun Lee.

Additional information

Mathematics Subject Classification (2000):14J45, 14N30.

The author was supported in part by KOSEF Grant R14-2002-007-01001-0(2002).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, S. Seshadri constants and Fano manifolds. Math. Z. 245, 645–656 (2003). https://doi.org/10.1007/s00209-003-0561-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0561-8

Keywords

Search

Navigation