Abstract
We characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of divisors computing minimal log discrepancies for two-dimensional varieties, which is a conjecture by Ishii and also a special case of the conjecture by Mustaţă–Nakamura.
Similar content being viewed by others
References
Artin, M.: Coverings of the rational double points in characteristic \(p\). In: Baily Jr., W.L., Shioda, T. (eds.) Complex Analysis and Algebraic Geometry, pp. 11–22. Cambridge University Press, Cambridge (1977)
de Fernex, T., Ein, L., Mustaţă, M.: Log canonical thresholds on varieties with bounded singularities. In: Faber, C., et al. (eds.) Classification of Algebraic Varieties. EMS Series of Congress Reports, pp. 221–257. European Mathematical Society, Zürich (2011)
Ein, L., Mustaţă, M.: Jet schemes and singularities. In: Abramovich, D. et al. (eds.) Algebraic Geometry—Seattle 2005. Proceedings of Symposia in Pure Mathematics, vol. 80.2, pp. 505–546. American Mathematical Society, Providence (2009)
Eisenbud, D.: Commutative Algebra. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)
Fedder, R.: \(F\)-purity and rational singularity. Trans. Amer. Math. Soc. 278(2), 461–480 (1983)
Hara, N., Watanabe, K.-I.: F-regular and F-pure rings vs. log terminal and log canonical singularities. J. Algebraic Geom. 11(2), 363–392 (2002)
Hirokado, M.: Deformation of rational double points and simple elliptic singularities in characteristic \(p\). Osaka J. Math. 41(3), 605–616 (2004)
Ishii, S.: Smoothness and jet schemes. In: Brasselet, J.-P. et al. (eds.) Singularities—Niigata–Toyama 2007. Advanced Studies in Pure Mathematics, vol. 56, pp. 187–199 (2009)
Ishii, S.: Introduction to Singularities, 2nd edn. Springer, Tokyo (2018)
Ishii, S.: Finite determination conjecture for Mather–Jacobian minimal log discrepancies and its applications. Eur. J. Math. 4(4), 1433–1475 (2018)
Ishii, S., Reguera, A.J.: Singularities in arbitrary characteristic via jet schemes. In: Ji, L. (ed.) Hodge Theory and \(L^2\)-Analysis. Advanced Lectures in Mathematics (ALM), vol. 39, pp. 419–449. International Press, Somerville (2017)
Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)
Kollár, J., Shepherd-Barron, N.I.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)
Kollár, J., Smith, K.E., Corti, A.: Rational and Nearly Rational Varieties. Cambridge Studies in Advanced Mathematics, vol. 92. Cambridge University Press, Cambridge (2004)
Liu, W., Rollenske, S.: Two-dimensional semi-log-canonical hypersurfaces. Matematiche (Catania) 67(2), 185–202 (2012)
Mustaţă, M., Nakamura, Y.: A boundedness conjecture for minimal log discrepancies on a fixed germ. In: Budur, N., et al. (eds.) Local and Global Methods in Algebraic Geometry. Contemporary Mathematics, vol. 712, pp. 287–306. American Mathematical Society, Providence (2018)
Acknowledgements
The author would like to thank Professor Shihoko Ishii and Professor Shunsuke Takagi for valuable conversations.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author is partially supported by JSPS Grant-in-Aid for Early-Career Scientists 19K14496 and the Iwanami Fujukai Foundation.
Rights and permissions
About this article
Cite this article
Shibata, K. Characterization of two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic. European Journal of Mathematics 7, 931–951 (2021). https://doi.org/10.1007/s40879-021-00484-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-021-00484-7