Skip to main content
Log in

The augmented base locus of real divisors over arbitrary fields

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We show that the augmented base locus coincides with the exceptional locus (i.e., null locus) for any nef \(\mathbb R\)-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in terms of the exceptional locus generalizing a result of Keel. We also discuss some problems related to augmented base loci of log divisors.

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

Access this article

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. Birkar, C: Existence of flips and minimal models for 3-folds in char p. Annales scientifiques de l’ENS 49, 169–212 (2016)

  2. Birkar, C., Hu, Z.: Log canonical pairs with good augmented base loci. Compos. Math. 150, 579–592 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boucksom, S., Cacciola, S., Lopez, A.F.: Augmented base loci and restricted volumes on normal varieties. Math. Z. 278, 979–985 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boucksom, S., Demailly, J.-P., Pǎun, M., Peternell, Th: The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension. J. Algebraic Geom. 22, 201–248 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cacciola, S., Lopez, A.F.: Nakamaye’s theorem on log canonical pairs. Ann Inst Fourier 64(6), 2283–2298 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cascini, P., McKernan, J., Mustaţă M.: The augmented base locus in positive characteristic. Proceedings of the Edinburgh Mathematical Society (series 2), 57(1), 79–87 (2014)

  7. Cascini, P., Tanaka, H., Xu, C.: On base point freeness in positive characteristic. Annales scientifiques de l’ENS 48, 1239–1272 (2015)

  8. Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M., Popa, M.: Restricted volumes and base loci of linear series. Amer. J. Math. 131(3), 607–651 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M., Popa, M.: Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56, 1701–1734 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fujino, O., Tanaka, H.: On log surfaces. Proc. Japan Acad. Ser. A Math. Sci. 88(8), 109–114 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fujita, T.: Vanishing theorems for semipositive line bundles. In: Algebraic geometry (Tokyo/Kyoto, 1982), 519–528. Lecture Notes Math. 1016, Springer, Berlin (1983)

  12. Keel, S.: Basepoint freeness for nef and big line bundles in positive characteristic. Ann Math Second Ser 149(1), 253–286 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lazarsfeld, R.: Positivity in algebraic geometry. I. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  14. Nakai, Y.: Some fundamental lemmas on projective schemes. Trans. Am. Math. Soc. 109, 296–302 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nakamaye, M.: Base loci of linear series are numerically determined. Trans. Am. Math. Soc. 355, 551–566 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nakamaye, M.: Stable base loci of linear series. Math. Ann. 318, 837–847 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shokurov, V. V.: Complements on surfaces. Algebraic Geom J. Math. Sci. (New York) 102(2), 3876–3932 (2000)

  18. The stacks project. http://stacks.math.columbia.edu

  19. Tanaka, H.: Minimal models and abundance for positive characteristic log surfaces. Nagoya Math. J. 216, 1–70 (2014)

  20. Xu, C.: On base point free theorem of threefolds in positive characteristic. To appear in J. Inst. Math. Jussieu. arXiv:1311.3819

Download references

Acknowledgments

I would like to thank Mircea Mustaţă and Karl Schwede for discussions related to Sect. 6, and thank the referee for the valuable comments and corrections. This work was supported by a grant of the Leverhulme Trust.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Caucher Birkar.

Additional information

Communicated by Ngaiming Mok.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Birkar, C. The augmented base locus of real divisors over arbitrary fields. Math. Ann. 368, 905–921 (2017). https://doi.org/10.1007/s00208-016-1441-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-016-1441-y

Mathematics Subject Classification

Navigation