Skip to main content
SpringerLink
Log in

A Simple Proof of ACC for Minimal Log Discrepancies for Surfaces

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Following Shokurov’s idea, we give a simple proof of the ACC conjecture for minimal log discrepancies for surfaces.

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. Alexeev, V.: Two two-dimensional terminations, Duke Math. J., 69(3), 527–545 (1993)

    Article  MathSciNet  Google Scholar 

  2. Ambro, F.: On minimal log discrepancies, Math. Res. Lett., 6, 573–580 (1999)

    Article  MathSciNet  Google Scholar 

  3. Ambro, F.: The set of toric minimal log discrepancies, Cent. Eur. J. Math., 4(3), 358–370 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  4. Borisov, A. A.: Minimal discrepancies of toric singularities, Manuscripta Math., 92(1), 33–45 (1997)

    Article  MathSciNet  Google Scholar 

  5. Birkar, C., Shokurov, V. V.: Mlds vs thresholds and flips, J. Reine Angew. Math., 638, 209–234 (2010)

    MathSciNet  Google Scholar 

  6. Chen, G., Han, J.: Boundedness of (ϵ, n)-complements for surfaces. arXiv:2002.02246v2. Short version published on, Adv. Math., 383, 107703, 40 pp. (2021)

    Article  Google Scholar 

  7. Chen, W., Gongyo, Y., Nakamura, Y.: On generalized minimal log discrepancy, arXiv:2112.09501v1

  8. Han, J., Li, Z., Qi, L.: ACC for log canonical threshold polytopes, Amer. J. Math., 143(3), 681–714 (2021)

    Article  MathSciNet  Google Scholar 

  9. Han, J., Liu, J., Luo, Y.: ACC for minimal log discrepancies of terminal threefolds, arXiv:2202.05287v2

  10. Han, J., Liu, J., Shokurov, V. V.: ACC for minimal log discrepancies of exceptional singularities, arXiv: 1903.04338v2

  11. Han, J., Liu, Y., Qi, L.: ACC for local volumes and boundedness of singularities. arXiv:2011.06509v2, to appear in J. Algebraic Geom.

  12. Han, J., Luo, Y.: On boundedness of divisors computing minimal log discrepancies for surfaces, to appear in J. Inst. Math. Jussieu, doi:https://doi.org/10.1017/S1474748022000299

  13. Jiang, C.: A gap theorem for minimal log discrepancies of non-canonical singularities in dimension three, J. Algebraic Geom., 30, 759–800 (2021)

    Article  MathSciNet  Google Scholar 

  14. Kawakita, M.: Discreteness of log discrepancies over log canonical triples on a fixed pair, J. Algebraic Geom., 23(4), 765–774 (2014)

    Article  MathSciNet  Google Scholar 

  15. Kawakita, M.: On equivalent conjectures for minimal log discrepancies on smooth threefolds, J. Algebraic Geom., 30, 97–149 (2021)

    Article  MathSciNet  Google Scholar 

  16. Kollár, J., Mori, S.: Birational geometry of algebraic varieties, Cambridge Tracts in Math., Vol. 134, Cambridge Univ. Press, Cambridge, 1998

    Book  Google Scholar 

  17. Lipman, J.: Desingularization of two-dimensional schemes, Ann. Math., 107(1), 151–207 (1978)

    Article  MathSciNet  Google Scholar 

  18. Liu, J.: Toward the equivalence of the ACC for a-log canonical thresholds and the ACC for minimal log discrepancies, arXiv:1809.04839v3

  19. Liu, J.: On Shokurov’s ascending chain condition conjecture for minimal log discrepancies, Thesis (Ph.D.)-The University of Utah, 2021

  20. Liu, J., Xiao, L.: An optimal gap of minimal log discrepancies of threefold non-canonical singularities. J. Pure Appl. Algebra, 225(9), 106674, 23 pp. (2021)

    Article  MathSciNet  Google Scholar 

  21. Liu, J., Xie, L.: Divisors computing minimal log discrepancies on lc surfaces, to appear in Math. Proc. Cambridge Philos. Soc, https://doi.org/10.1017/S0305004123000051

  22. Mustaţă, M., Nakamura, Y.: A boundedness conjecture for minimal log discrepancies on a fixed germ, In: Local and Global Methods in Algebraic Geometry. Contemp. Math., Vol. 712, AMS, Providence, 287–306, 2018

    Chapter  Google Scholar 

  23. Nakamura, Y., Shibata, K.: Inversion of adjunction for quotient singularities, arXiv:2011.07300, to appear in Algebr. Geom.

  24. Nakamura, Y., Shibata, K.: Inversion of adjunction for quotient singularities II: Non-linear actions, arXiv:2112.09502v1

  25. Shokurov, V. V.: Problems about Fano varieties, In: Birational Geometry of Algebraic Varieties, Open Problems. The XXIIIrd International Symposium, Division of Mathematics, The Taniguchi Foundation, 30–32, August 22–August 27, 1988

  26. Shokurov, V. V.: A.c.c. in codimension 2 (preprint), 1994

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

    Article  MathSciNet  Google Scholar 

  28. Shokurov, V. V.: Letters of a bi-rationalist, V. Minimal log discrepancies and termination of log flips (Russian), Tr. Mat. Inst. Steklova, 246, Algebr. Geom. Metody, Svyazi i Prilozh., 328–351, 2004

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Tanaka, H.: The X-method for klt surfaces in positive characteristic, J. Algebraic Geom., 24(4), 605–628 (2015)

    Article  MathSciNet  Google Scholar 

  31. Tanaka, H.: Minimal model program for excellent surfaces. Ann. Inst. Fourier (Grenoble), 68(1), 345–376 (2018)

    Article  MathSciNet  CAS  Google Scholar 

  32. Zhuang, Z.: On boundedness of singularities and minimal log discrepancies of Kollár components, arXiv: 2202.06455v1

Download references

Acknowledgements

We would like to express our deepest gratitude to Vyacheslav Shokurov who not only shared us with his preprint [26] but also shaped our research styles. The draft of this paper was written during the reading seminar in birational geometry at Johns Hopkins University in Spring 2019. We would like to thank Yang He and Xiangze Zeng for joining in the seminar and useful discussions. We would like to thank the referees for their comments and for suggesting us some references on mlds.

Author information

Authors and Affiliations

  1. Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, Shanghai, 200438, P. R. China

    Jing Jun Han

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Yu Jie Luo

Authors
  1. Jing Jun Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Jie Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu Jie Luo.

Ethics declarations

Conflict of Interest The authors declare no conflict of interest.

Additional information

The first named author was supported by National Key Research and Development Program of China (Grant No. 2020YFA0713200), The first named author is a member of LMNS, Fudan University

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, J.J., Luo, Y.J. A Simple Proof of ACC for Minimal Log Discrepancies for Surfaces. Acta. Math. Sin.-English Ser. 40, 425–434 (2024). https://doi.org/10.1007/s10114-023-2094-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-023-2094-x

Keywords

MR(2010) Subject Classification

Search

Navigation