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ON DEL PEZZO FIBRATIONS IN POSITIVE CHARACTERISTIC

Part of: Birational geometry Surfaces and higher-dimensional varieties Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  30 March 2020

Fabio Bernasconi Open the ORCID record for Fabio Bernasconi [Opens in a new window]  and
Hiromu Tanaka
Fabio Bernasconi
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA (fabio@math.utah.edu)
Hiromu Tanaka
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo153-8914, Japan (tanaka@ms.u-tokyo.ac.jp)
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Abstract

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.

Keywords

minimal model programgeneric fibresdel Pezzo surfacespositive characteristic

MSC classification

Primary: 14E30: Minimal model program (Mori theory, extremal rays) 14G17: Positive characteristic ground fields 14J45: Fano varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 1 , January 2022 , pp. 197 - 239
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Artin, M., Coverings of the rational double points in characteristic p , in Complex Analysis and Algebraic Geometry, pp. 11–22 (Iwanami Shoten, Tokyo, 1977).CrossRefGoogle Scholar
Becker, M. F. and MacLane, S., The minimum number of generators for inseparable algebraic extensions, Bull. Amer. Math. Soc. (N.S.) 46 (1940), 182–186.CrossRefGoogle Scholar
Bernasconi, F., Kawamata–Viehweg vanishing fails for log del Pezzo surfaces in char. 3, preprint, 2019, arXiv:1709.09238.Google Scholar
Birkar, C. and Waldron, J., Existence of Mori fibre spaces for 3-folds in char p , Adv. Math. 313 (2017), 62–101.CrossRefGoogle Scholar
Cascini, P. and Tanaka, H., Smooth rational surfaces violating Kawamata–Viehweg vanishing, Eur. J. Math. 4(1) (2018), 162–176.CrossRefGoogle Scholar
Cascini, P. and Tanaka, H., Purely log terminal threefolds with non-normal centres in characteristic two, Amer. J. Math. 141(4) (2019), 941–979.CrossRefGoogle Scholar
Cascini, P., Tanaka, H. and Witaszek, J., On log del Pezzo surfaces in large characteristic, Compos. Math. 153(4) (2017), 820–850.CrossRefGoogle Scholar
Das, O., Kawamata-Viehweg Vanishing Theorem for del Pezzo Surfaces over imperfect fields in characteristic $p>3$ , preprint, 2019, arXiv:1709.03237.3$+,+preprint,+2019,+arXiv:1709.03237.>Google Scholar
Dolgachev, I. V., Classical Algebraic Geometry. A Modern View (Cambridge University Press, Cambridge, 2012).CrossRefGoogle Scholar
Ejiri, S., Positivity of anti-canonical divisors and F-purity of fibers, Algebra Number Theory 13(9) (2019), 2057–2080.CrossRefGoogle Scholar
Fanelli, A. and Schröer, S., Del Pezzo surfaces and Mori fiber spaces in positive characteristic, Trans. Amer. Math. Soc. 373(3) (2020), 1775–1843.CrossRefGoogle Scholar
Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S. L., Nitsure, N. and Vistoli, A., Fundamental Algebraic Geometry. Grothendieck’s FGA Explained, Mathematical Surveys and Monographs, Volume 123 (American Mathematical Society, Providence, RI, 2005).Google Scholar
Fu, L., Etale Cohomology Theory, Revised edition, Nankai Tracts in Mathematics, Volume 14 (World Scientific Publishing Co. Pte. Ltd, Hackensack, NJ, 2015).CrossRefGoogle Scholar
Fujino, O. and Gongyo, Y., On canonical bundle formulas and subadjunctions, Michigan Math. J. 60(3) (2012), 255–264.Google Scholar
Gongyo, Y., Nakamura, Y. and Tanaka, H., Rational points on log Fano threefolds over a finite field, J. Eur. Math. Soc. (JEMS) 21 (2019), 3759–3795.CrossRefGoogle Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, Volume 52 (Springer-Verlag, NewYork, 1977).CrossRefGoogle Scholar
Huneke, C. and Swanson, I., Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, Volume 336 (Cambridge University Press, Cambridge, 2006).Google Scholar
Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model program, in Algebraic Geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, Volume 10, pp. 283–360 (North-Holland, Amsterdam, 1987).CrossRefGoogle Scholar
Kollár, J., Rational curves on algebraic varieties, in Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics (Springer-Verlag, Berlin, 1996).Google Scholar
Kollár, J., Singularities of the Minimal Model Program, With a collaboration of Sándor Kovács. Cambridge Tracts in Mathematics, Volume 200 (Cambridge University Press, Cambridge, 2013).CrossRefGoogle Scholar
Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, Volume 134 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
Lang, S., On quasi algebraic closure, Ann. of Math. (2) 2(55) (1952), 373–390.CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 49 (Springer-Verlag, Berlin, 2004).Google Scholar
Lipman, J., Desingularization of two-dimensional schemes, Ann. of Math. (2) 107(1) (1978), 151–207.CrossRefGoogle Scholar
Liu, Q., Algebraic Geometry and Arithmetic Curves, Translated from the French by Reinie Erne. Oxford Graduate Texts in Mathematics, Volume 6 (Oxford Science Publications. Oxford University Press, Oxford, 2002).Google Scholar
Maddock, Z., Regular del Pezzo surfaces with irregularity, J. Algebraic Geom. 25(3) (2016), 401–429.CrossRefGoogle Scholar
Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, Volume 8 (Cambridge University Press, Cambridge, 1989). Translated from the Japanese by M. Reid.Google Scholar
Mukai, S., Counterexamples to Kodaira’s vanishing and Yau’s inequality in positive characteristic, Kyoto J. Math. 53(2) (2013), 515–532.CrossRefGoogle Scholar
Nakamura, Y. and Tanaka, H., A Witt Nadel vanishing theorem, Compos. Math. 156(3) (2020), 435–475.CrossRefGoogle Scholar
Patakfalvi, Z. and Waldron, J., Singularities of general fibers and the LMMP, preprint, 2019, arXiv:1708.04268v2.Google Scholar
Reichstein, Z. and Youssin, B., Essential dimensions of algebraic groups and a resolution theorem for G-varieties. With an appendix by János Kollár and Endre Szabó, Canad. J. Math. 52(5) (2000), 1018–1056.CrossRefGoogle Scholar
Reid, M., Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30(5) (1994), 695–727.CrossRefGoogle Scholar
Schröer, S., Weak del Pezzo surfaces with irregularity, Tohoku Math. J. 59(2) (2007), 293–322.CrossRefGoogle Scholar
Schröer, S., Singularities appearing on generic fibers of morphisms between smooth schemes, Michigan Math. J. 56(1) (2008), 55–76.CrossRefGoogle Scholar
Schröer, S., On fibrations whose geometric fibers are nonreduced, Nagoya Math. J. 200 (2010), 35–57.CrossRefGoogle Scholar
The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu.Google Scholar
Tanaka, H., Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. 216 (2014), 1–70.CrossRefGoogle Scholar
Tanaka, H., The X-method for klt surfaces in positive characteristic, J. Algebraic Geom. 24(4) (2015), 605–628.CrossRefGoogle Scholar
Tanaka, H., Minimal model program for excellent surfaces, Ann. Inst. Fourier. (Grenoble) 68(1) (2018), 345–376.CrossRefGoogle Scholar
Tanaka, H., Behavior of canonical divisors under purely inseparable base changes, J. Reine Angew. Math. 744 (2018), 237–264.Google Scholar
Tanaka, H., Pathologies on Mori fibre spaces in positive characteristic, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), to appear, preprint, arXiv:1609.00574v3.Google Scholar
Tanaka, H., Invariants of algebraic varieties over imperfect fields, preprint, 2019,arXiv:1903.10113v2.Google Scholar