The minimal log discrepancies on a smooth surface in positive characteristic


This paper shows that Mustaţǎ–Nakamura’s conjecture holds for pairs consisting of a smooth surface and a multiideal with a real exponent over the base field of positive characteristic. As corollaries, we obtain the ascending chain condition of the minimal log discrepancies and of the log canonical thresholds for those pairs. We also obtain finiteness of the set of the minimal log discrepancies of those pairs for a fixed real exponent.

This is a preview of subscription content, access via your institution.


  1. 1.

    Birkar, C.: Existence of flips and minimal models for 3-folds in char \(p\). Ann. Scient. L’École Norm. Sup. 49, 169–212 (2016)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Hacon, C., McKernan, J., Xu, C.: ACC for log canonical thresholds. Ann. Math. 180, 523–571 (2014)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ishii, S.: Finite determination conjecture for Mather–Jacobian minimal log discrepancies and its applications. Eur. J. Math. 4, 1433–1475 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ishii, S.: Inversion of modulo \(p\) reduction and a partial descent from characteristic \(0\) to positive characteristic, preprint (2018). arXiv:1808.10155, to appear in the Proceedings of JARCS VII,

  5. 5.

    Ishii, S., Reguera, A.: Singularities in arbitrary characteristic via jet schemes. Math. Zeitschrift 275(3–4), 1255–1274 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

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

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kawakita, M.: Divisors computing the minimal log discrepancy on a smooth surface. Math. Proc. Camb. Philos. Soc. 163(1), 187–192 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Kawakita, M.: On equivalent conjectures for minimal log discrepancies on smooth threefolds. arXiv:1803.02539

  9. 9.

    Kollár, J., Smith, K., Corti, A.: Rational and nearly rational varieties. Camb. Stud. Adv. Math. 92, 235 (2002)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Mustaţǎ, M., Nakamura, Y.: A boundedness conjecture for minimal log discrepancies on a fixed germ. AMS Contemp. Math. 712, 287–306 (2018)

    MathSciNet  Article  Google Scholar 

Download references


The author expresses her hearty thanks to Kohsuke Shibata for his insightful comments which improves the paper. She also would like to thank Masayuki Kawakita, Lawrence Ein and Mircea Mustaţǎ for the useful discussions. A big part of these discussions was done during the author’s stay in MSRI (Program: Birational Geometry and Moduli Theory) and she is grateful for the support of MSRI. The author would like to thank the referee for useful comments to improve the paper.

Author information



Corresponding author

Correspondence to Shihoko Ishii.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

S. Ishii: The author is partially supported by JSPS 19K03428.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ishii, S. The minimal log discrepancies on a smooth surface in positive characteristic. Math. Z. 297, 389–397 (2021).

Download citation