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On log del Pezzo surfaces in large characteristic

  • Paolo Cascini (a1), Hiromu Tanaka (a2) and Jakub Witaszek (a3)
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.

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[Ale93] Alexeev V., Two two-dimensional terminations , Duke Math. J. 69 (1993), 527545.
[Ale94] Alexeev V., Boundedness and K 2 for log surfaces , Int. J. Math. 5 (1994), 779810.
[Băd01] Bădescu L., Algebraic surfaces (Universitext, Springer, New York, 2001).
[Bir16] Birkar C., Existence of flips and minimal models for 3-folds in char p , Ann. Sci. Éc. Norm. Supér. 49 (2016), 169212.
[BW14] Birkar C. and Waldron J., Existence of Mori fibre spaces for 3-folds in char inline-graphic $p$ , Preprint (2014), arXiv:1410.4511.
[CGS16] Cascini P., Gongyo Y. and Schwede K., Uniform bounds for strongly F-regular surfaces , Trans. Amer. Math. Soc. 368 (2016), 55475563.
[CTW17] 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.
[CTX15] Cascini P., Tanaka H. and Xu C., On base point freeness in positive characteristic , Ann. Sci. Éc. Norm. Supér. 48 (2015), 12391272.
[Das15] Das O., On strongly F-regular inversion of adjunction , J. Algebra 434 (2015), 207226.
[EV92] Esnault H. and Viehweg E., Lectures on vanishing theorems, DMV Seminar, vol. 20 (Birkhäuser, Basel, 1992).
[FGIKNV05] Fantechi B., Göttsche L., Illusie L., Kleiman S. L., Nitsure N. and Vistoli A., Fundamental algebraic geometry (American Mathematical Society, Providence, RI, 2005).
[FK88] 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).
[Fu15] Fu L., Etale cohomology theory, Nankai Tracts in Mathematics, vol. 14, revised edn (World Scientific, Hackensack, NJ, 2015).
[Fuj12] Fujino O., Minimal model theory for log surfaces , Publ. Res. Inst. Math. Sci. 48 (2012), 339371.
[HX15] Hacon C. and Xu C., On the three dimensional minimal model program in positive characteristic , J. Amer. Math. Soc. 28 (2015), 711744.
[Har98] Hara N., A characterization of rational singularities in terms of injectivity of Frobenius maps , Amer. J. Math. 120 (1998), 981996.
[KM99] Keel S. and McKernan J., Rational curves on quasi-projective surfaces, Memoirs American Mathematical Society, vol. 669 (American Mathematical Society, Providence, RI, 1999).
[Kol13] Kollár J., Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200 (Cambridge University Press, Cambridge, 2013).
[KM98] Kollár J. and Mori S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).
[Kol92] Kollár J. et al. , Flips and abundance for algebraic threefolds (Société Mathématique de France, Paris, 1992).
[Lan16] Langer A., The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic , Duke Math. J. 165 (2016), 27372769.
[Liu02] Liu Q., Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford University Press, Oxford, 2002).
[MP04] McKernan J. and Prokhorov Y., Threefold thresholds , Manuscripta Math. 114 (2004), 281304.
[Pro01] Prokhorov Y., Lectures on complements on log surfaces, MSJ Memoirs, vol. 10 (Mathematical Society of Japan, Tokyo, 2001).
[Sch09] Schwede K., F-adjunction , Algebra Number Theory 3 (2009), 907950.
[Sch14] Schwede K., A canonical linear system associated to adjoint divisors in characteristic p > 0 , J. Reine Angew. Math. 696 (2014), 6987.
[SS10] Schwede K. and Smith K. E., Globally F-regular and log Fano varieties , Adv. Math. 224 (2010), 863894.
[Tan14] Tanaka H., Minimal models and abundance for positive characteristic log surfaces , Nagoya Math. J. 216 (2014), 170.
[Tan15] Tanaka H., The X-method for klt surfaces in positive characteristic , J. Algebraic Geom. 24 (2015), 605628.
[Wat91] Watanabe K., F-regular and F-pure normal graded rings , J. Pure Appl. Algebra 71 (1991), 341350.
[Wit15] Witaszek J., Effective bounds on singular surfaces in positive characteristic , Michigan Math. J. (2015), to appear.
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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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