Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-19T00:52:10.875Z Has data issue: false hasContentIssue false

GENERAL HYPERPLANE SECTIONS OF THREEFOLDS IN POSITIVE CHARACTERISTIC

Published online by Cambridge University Press:  12 April 2018

Kenta Sato
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan (ktsato@ms.u-tokyo.ac.jp)
Shunsuke Takagi
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan (stakagi@ms.u-tokyo.ac.jp)

Abstract

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

Type
Research Article
Copyright
© Cambridge University Press 2018

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.)

References

Aoyama, Y., Some basic results on canonical modules, J. Math. Kyoto. Univ. 23 (1983), 8594.Google Scholar
Artin, M., On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129136.Google Scholar
Birkar, C., Existence of flips and minimal models for 3-folds in char p, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), 169212.Google Scholar
Birkar, C. and Waldron, J., Existence of Mori fibre spaces for 3-folds in char p, Adv. Math. 313 (2017), 62101.Google Scholar
Cascini, P., Gongyo, Y. and Schwede, K., Uniform bounds for strongly F-regular surfaces, Trans. Amer. Math. Soc. 368 (2016), 55475563.Google Scholar
Cossart, V. and Piltant, O., Resolution of singularities of threefolds in positive characteristic. I, J. Algebra 320 (2008), 10511082.Google Scholar
Cossart, V. and Piltant, O., Resolution of singularities of threefolds in positive characteristic. II, J. Algebra 321 (2009), 18361976.Google Scholar
de Fernex, T. and Docampo, R., Jacobian discrepancies and rational singularities, J. Eur. Math. Soc. 16 (2014), 165199.Google Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Bounds for log canonical thresholds with applications to birational geometry, Math. Res. Lett. 10 (2003), 219236.Google Scholar
Ein, L. and Ishii, S., Singularities with respect to Mather–Jacobian discrepancies, in Commutative Algebra and Noncommutative Algebraic Geometry, Vol. II, Mathematical Sciences Research Institute Publications, Volume 68, pp. 125168 (Cambridge Univ. Press, New York, 2015).Google Scholar
Fulton, W., Intersection Theory, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 2 (Springer, Berlin, 1998).Google Scholar
Gabber, O., Notes on some t-structures, in Geometric Aspects of Dwork Theory, Vol. I, II, pp. 711734 (Walter de Gruyter GmbH & Co. KG, Berlin, 2004).Google Scholar
Greco, S. and Marinari, M. G., Nagata’s criterion and openness of loci for Gorenstein and complete intersection, Math. Z. 160 (1978), 207216.Google Scholar
Hacon, C. and Xu, C., On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), 711744.Google Scholar
Hara, N., Classification of two-dimensional F-regular and F-pure singularities, Adv. Math. 133 (1998), 3353.Google Scholar
Hara, N. and Takagi, S., On a generalization of test ideals, Nagoya Math. J. 175 (2004), 5974.Google Scholar
Hara, N. and Watanabe, K.-i., F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebra Geom. 11 (2002), 363392.Google Scholar
Hartshorne, R., Residues and Duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64, with an appendix by P. Deligne, Lecture Notes in Mathematics, Volume 20 (Springer-Verlag, Berlin–New York, 1966).Google Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, Volume 52, (Springer-Verlag, Berlin–New York, 1977).Google Scholar
Hirokado, M., Canonical singularities of dimension three in characteristic 2 which do not follow Reid’s rules, Kyoto J. Math. (to appear) arXiv:1607.08664.Google Scholar
Hirokado, M., Ito, H. and Saito, N., Three dimensional canonical singularities in codimension two in positive characteristic, J. Algebra 373 (2013), 207222.Google Scholar
Hashizume, K., Nakamura, Y. and Tanaka, H., Minimal model program for log canonical threefolds in positive characteristic, preprint, arXiv:1711.10706.Google Scholar
Hochster, M. and Huneke, C., Tight closure and strong F-regularity, Mém. Soc. Math. France 38 (1989), 119133.Google Scholar
Ishii, S. and Reguera, J., Singularities in arbitrary characteristic via jet schemes, in Hodge Theory and L 2 Analysis, Advanced Lectures in Mathematics (ALM), Volume 39, pp. 419449 (International Press, Somerville, MA, 2017). Higher Education Press, Beijing.Google Scholar
Kollár, J., Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics, Volume 200 (Cambridge University Press, Cambridge, 2013).Google Scholar
Kunz, E., On Noetherian rings of characteristic p, Amer. J. Math. 98 (1976), 9991013.Google Scholar
Lipman, J., Rational singularities, with applications to algebraic surfaces and unique factorization, Publications Mathématiques. Institut de Hautes Études Scientifiques, Volume 36, pp. 195279. (1969).Google Scholar
McKernan, J. and Prokhorov, Y., Threefold thresholds, Manuscripta Math. 114 (2004), 281304.Google Scholar
Reid, M., Canonical 3-folds, in Journées de géométrie algébrique d’Angers 1979 (ed. Beauville, A.), pp. 273310 (Sijthoff & Noordhoff, Alphen, 1980).Google Scholar
Schwede, K. and Smith, K. E., Globally F-regular and log Fano varieties, Adv. Math. 224 (2010), 863894.Google Scholar
Schwede, K. and Tucker, K., On the behavior of test ideals under finite morphisms, J. Algebraic. Geom. 23 (2014), 399443.Google Scholar
Schwede, K. and Zhang, W., Bertini theorems for F-singularities, Proc. Lond. Math. Soc. 107 (2013), 851874.Google Scholar
Tanaka, H., Minimal model program for excellent surfaces, Ann. Inst. Fourier. (to appear) arXiv:1608.07676.Google Scholar
Takagi, S. and Watanabe, K.-i., F-singularities: applications of characteristic p methods to singularity theory, Sugaku Expositions 31 (2018), 142.Google Scholar
Watanabe, K., Plurigenera of normal isolated singularities. I, Math. Ann. 250 (1980), 6594.Google Scholar
Watanabe, K.-i., F-regular and F-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), 341350.Google Scholar