Hostname: page-component-5b777bbd6c-f9nfp Total loading time: 0 Render date: 2025-06-19T00:12:08.961Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic singularities and threefold flops in positive characteristic

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  27 May 2025

Hiromu Tanaka
Hiromu Tanaka*
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan
*
Email: tanaka.hiromu.7z@kyoto-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of X is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities defined over imperfect fields.

Keywords

elliptic singularitiesthree-dimensionalterminal singularitiesflopspositive characteristic

MSC classification

Primary: 14J17: Singularities
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 57
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bădescu, L.. Algebraic Surfaces. Universitext, Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author.10.1007/978-1-4757-3512-3CrossRefGoogle Scholar
Bourbaki, N.. Commutative Algebra. Chapters 1–7. Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French, Reprint of the 1972 edition.Google Scholar
Cascini, P., Tanaka, H., and Xu, C.. On base point freeness in positive characteristic. Ann. Sci. Éc. Norm. Supér. (4), 48(5): (2015), 1239–1272.10.24033/asens.2269CrossRefGoogle Scholar
Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S. L., Nitsure, N., and Vistoli, A.. Fundamental algebraic geometry. Mathematical Surveys and Monographs, Vol. 123. American Mathematical Society, Providence, RI, 2005. Grothendieck’s FGA explained.Google Scholar
Fujita, T.. Classification theories of polarized varieties. London Mathematical Society Lecture Note Series, Vol. 155. Cambridge University Press, Cambridge, 1990.Google Scholar
Hartshorne, R.. Algebraic geometry. Graduate Texts in Mathematics, 52. Springer-Verlag, New York-Heidelberg, 1977.Google Scholar
Ishii, S.. Introduction to Singularities. Springer, Tokyo, 2018. Second edition of [MR3288750].10.1007/978-4-431-56837-7CrossRefGoogle Scholar
Kollár, J. and Mori, S.. Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, Vol. 134. Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.Google Scholar
Kollár, J.. Singularities of the minimal model program. Cambridge Tracts in Mathematics, Vol. 200. Cambridge University Press, Cambridge, 2013. With a collaboration of S. Kovács.Google Scholar
Kollár, J.. Flops. Nagoya Math. J., 113: (1989), 15–36.10.1017/S0027763000001240CrossRefGoogle Scholar
Laufer, H. B.. On minimally elliptic singularities. Amer. J. Math., 99(6): (1977), 1257–1295.10.2307/2374025CrossRefGoogle Scholar
Lipman, J.. Rational singularities, with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math., 36: (1969), 195–279.10.1007/BF02684604CrossRefGoogle Scholar
Matsumura, H.. Commutative ring theory. Cambridge Studies in Advanced Mathematics, Vol. 8. Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid.Google Scholar
Nagata, M.. Local rings. Interscience Tracts in Pure and Applied Mathematics, 13. Interscience Publishers, New York-London, 1962.Google Scholar
Schröer, S.. Singularities appearing on generic fibers of morphisms between smooth schemes. Michigan Math. J., 56(1): (2008), 55–76.10.1307/mmj/1213972397CrossRefGoogle Scholar
Schröer, S.. The structure of regular genus-one curves over imperfect fields. Preprint available at arXiv:2211.04073v1, 2022.Google Scholar
Schwede, K. and Smith, K. E.. Globally F-regular and log Fano varieties. Adv. Math., 224(3): (2010), 863–894.10.1016/j.aim.2009.12.020CrossRefGoogle Scholar
Tanaka, H.. The X-method for klt surfaces in positive characteristic. J. Algebraic Geom., 24(4): (2015), 605–628.10.1090/S1056-3911-2014-00627-5CrossRefGoogle Scholar
Tanaka, H.. Behavior of canonical divisors under purely inseparable base changes. J. Reine Angew. Math., 744: (2018), 237–264.10.1515/crelle-2015-0111CrossRefGoogle Scholar
Tanaka, H.. Minimal model program for excellent surfaces. Ann. Inst. Fourier (Grenoble), 68(1): (2018), 345–376.10.5802/aif.3163CrossRefGoogle Scholar
Tanaka, H.. Invariants of algebraic varieties over imperfect fields. Tohoku Math. J. (2), 73(4): (2021), 471–538.10.2748/tmj.20200611CrossRefGoogle Scholar
Tanaka, H.. Fano threefolds in positive characteristic II. Preprint available at arXiv:2308.08122, to appear in Kyoto J. Math., 2023.Google Scholar
Tanaka, H.. Fano threefolds in positive characteristic IV. Preprint available at arXiv:2308.08127, 2023.Google Scholar
Tanaka, H.. Bertini theorems admitting base changes. J. Algebra, 644: (2024), 64–125.10.1016/j.jalgebra.2023.12.038CrossRefGoogle Scholar
Wagreich, P.. Elliptic singularities of surfaces. Amer. J. Math., 92: (1970), 419–454.10.2307/2373333CrossRefGoogle Scholar