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On log del Pezzo surfaces in large characteristic

Part of: General commutative ring theory (Co)homology theory Birational geometry

Published online by Cambridge University Press:  08 March 2017

Paolo Cascini ,
Hiromu Tanaka  and
Jakub Witaszek
Paolo Cascini
Affiliation:
Department of Mathematics, Imperial College, London, 180 Queen’s Gate, London SW7 2AZ, UK email [email protected]
Hiromu Tanaka
Affiliation:
Department of Mathematics, Imperial College, London, 180 Queen’s Gate, London SW7 2AZ, UK email [email protected]
Jakub Witaszek
Affiliation:
Department of Mathematics, Imperial College, London, 180 Queen’s Gate, London SW7 2AZ, UK email [email protected]
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Abstract

We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally $F$-regular or it admits a log resolution which lifts to characteristic zero. As a consequence, we prove the Kawamata–Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.

Keywords

log del Pezzo surfaceF-singularitiesKawamata–Viehweg vanishingpositive characteristic

MSC classification

Primary: 14E30: Minimal model program (Mori theory, extremal rays) 14F17: Vanishing theorems 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 4 , April 2017 , pp. 820 - 850
Copyright
© The Authors 2017 

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References

Alexeev, V., Two two-dimensional terminations , Duke Math. J. 69 (1993), 527–545.CrossRefGoogle Scholar
Alexeev, V., Boundedness and K 2 for log surfaces , Int. J. Math. 5 (1994), 779–810.Google Scholar
Bădescu, L., Algebraic surfaces (Universitext, Springer, New York, 2001).CrossRefGoogle Scholar
Birkar, C., Existence of flips and minimal models for 3-folds in char p , Ann. Sci. Éc. Norm. Supér. 49 (2016), 169–212.CrossRefGoogle Scholar
Birkar, C. and Waldron, J., Existence of Mori fibre spaces for 3-folds in char $p$ , Preprint (2014), arXiv:1410.4511.Google Scholar
Cascini, P., Gongyo, Y. and Schwede, K., Uniform bounds for strongly F-regular surfaces , Trans. Amer. Math. Soc. 368 (2016), 5547–5563.Google Scholar
Cascini, P., Tanaka, H. and Witaszek, J., Klt del Pezzo surfaces which are not globally F-split , Int. Math. Res. Not. IMRN (2017), doi:10.1093/imrn/rnw300.Google Scholar
Cascini, P., Tanaka, H. and Xu, C., On base point freeness in positive characteristic , Ann. Sci. Éc. Norm. Supér. 48 (2015), 1239–1272.Google Scholar
Das, O., On strongly F-regular inversion of adjunction , J. Algebra 434 (2015), 207–226.CrossRefGoogle Scholar
Esnault, H. and Viehweg, E., Lectures on vanishing theorems, DMV Seminar, vol. 20 (Birkhäuser, Basel, 1992).CrossRefGoogle Scholar
Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S. L., Nitsure, N. and Vistoli, A., Fundamental algebraic geometry (American Mathematical Society, Providence, RI, 2005).Google Scholar
Freitag, E. and Kiehl, R., Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 13 (Springer, Berlin, 1988).CrossRefGoogle Scholar
Fu, L., Etale cohomology theory, Nankai Tracts in Mathematics, vol. 14, revised edn (World Scientific, Hackensack, NJ, 2015).CrossRefGoogle Scholar
Fujino, O., Minimal model theory for log surfaces , Publ. Res. Inst. Math. Sci. 48 (2012), 339–371.CrossRefGoogle Scholar
Hacon, C. and Xu, C., On the three dimensional minimal model program in positive characteristic , J. Amer. Math. Soc. 28 (2015), 711–744.Google Scholar
Hara, N., A characterization of rational singularities in terms of injectivity of Frobenius maps , Amer. J. Math. 120 (1998), 981–996.Google Scholar
Keel, S. and McKernan, J., Rational curves on quasi-projective surfaces, Memoirs American Mathematical Society, vol. 669 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Kollár, J., Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200 (Cambridge University Press, Cambridge, 2013).Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
Kollár, J. et al. , Flips and abundance for algebraic threefolds (Société Mathématique de France, Paris, 1992).Google Scholar
Langer, A., The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic , Duke Math. J. 165 (2016), 2737–2769.Google Scholar
Liu, Q., Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford University Press, Oxford, 2002).Google Scholar
McKernan, J. and Prokhorov, Y., Threefold thresholds , Manuscripta Math. 114 (2004), 281–304.CrossRefGoogle Scholar
Prokhorov, Y., Lectures on complements on log surfaces, MSJ Memoirs, vol. 10 (Mathematical Society of Japan, Tokyo, 2001).Google Scholar
Schwede, K., F-adjunction , Algebra Number Theory 3 (2009), 907–950.Google Scholar
Schwede, K., A canonical linear system associated to adjoint divisors in characteristic p > 0 , J. Reine Angew. Math. 696 (2014), 69–87.CrossRefGoogle Scholar
Schwede, K. and Smith, K. E., Globally F-regular and log Fano varieties , Adv. Math. 224 (2010), 863–894.CrossRefGoogle 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 (2015), 605–628.Google Scholar
Watanabe, K., F-regular and F-pure normal graded rings , J. Pure Appl. Algebra 71 (1991), 341–350.Google Scholar
Witaszek, J., Effective bounds on singular surfaces in positive characteristic , Michigan Math. J. (2015), to appear.Google Scholar