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Singularities of Normal Log Canonical del Pezzo Surfaces of Rank One

Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 319)

Abstract

Let X be a normal del Pezzo surface of rank one with only rational log canonical singular points. In this paper, we prove that X can have at most one non klt singular point.

Keywords

Normal del Pezzo surface Log canonical singular point klt singular point Logarithmic Kodaira dimension 

2010 Mathematics Subject Classification

Primary 14J26 Secondary 14J17 14R25 

References

  1. 1.
    Alexeev, V.: Classification of log canonical surface singularities: arithmetical proof. Flips and abundance for algebraic threefolds. Kollár, J. et al.: Astérisque 211, 47–58 (1992). (Société Mathématique de France)Google Scholar
  2. 2.
    Alexeev, V., Nikulin, V.V.: Del Pezzo and  \(K3\)  Surfaces. MSJ Memoirs, vol. 15. Mathematical Society of Japan (2006)Google Scholar
  3. 3.
    Artin, M.: Some numerical criteria for contractability of curves on algebraic surfaces. Am. J. Math. 84, 485–496 (1962)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cheltsov, I.: Del Pezzo surfaces with nonrational singularities. Math. Notes 62, 377–389 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fujino, O.: Minimal model theory for log surfaces. Publ. Res. Inst. Math. Sci. 48, 339–371 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fujino, O., Tanaka, H.: On log surfaces. Proc. Jpn. Acad. Ser. A Math. Sci. 88, 109–114 (2012)Google Scholar
  7. 7.
    Fujita, K.: Log del Pezzo surfaces with large volumes. Kyushu J. Math. 70, 131–147 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fujita, K.: Log del Pezzo surfaces with not small fractional indices. Math. Nachr. 289, 34–59 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fujita, K., Yasutake, K.: Classification of log del Pezzo surfaces of index three. J. Math. Soc. Jpn. 69, 163–225 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Furushima, M.: Singular del Pezzo surfaces and analytic compactifications of \(3\)-dimensional complex affine space \(\mathbb{C}^3\). Nagoya Math. J. 104, 1–28 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hidaka, F., Watanabe, K.: Normal Gorenstein surfaces with ample anti-canonical divisor. Tokyo J. Math. 4, 319–330 (1981)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kojima, H.: On normal surfaces with strictly nef anticanonical divisors. Arch. Math. 77, 517–521 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kojima, H.: Rational unicuspidal curves on \(\mathbb{Q}\)-homology planes whose complements have logarithmic Kodaira dimension \(-\infty \). Nihonkai Math. J. 29, 29–43 (2018)Google Scholar
  14. 14.
    Kojima, H.: Some results on open algebraic surfaces of logarithmic Kodaira dimension zero. J. Algebra 547, 238–261 (2020)Google Scholar
  15. 15.
    Miyanishi, M.: Singularities of normal affine surfaces containing cylinderlike open sets. J. Algebra 68, 268–275 (1981)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Miyanishi, M.: Open Algebraic Surfaces. CRM Monograph Series, vol. 12. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  17. 17.
    Miyanishi, M., Tsunoda, S.: Non-complete algebraic surfaces with logarithmic Kodaira dimension \(-\infty \) and with non-connected boundaries at infinity. Jpn. J. Math. 10, 195–242 (1984)CrossRefGoogle Scholar
  18. 18.
    Mumford, D.: The topology of normal surface singularities of an algebraic surface and a criterion for simplicity. Publ. IHES 9, 5–22 (1961)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nakayama, N.: Classification of log del Pezzo surfaces of index two. J. Math. Sci. Univ. Tokyo 14, 293–498 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sakai, F.: Weil divisors on normal surfaces. Duke Math. J. 51, 877–887 (1984)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schröer, S.: On contractible curves on normal surfaces. J. Reine Angew. Math. 524, 1–15 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schröer, S.: Normal del Pezzo surfaces containing a nonrational singularity. Manuscripta Math. 104, 257–274 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tanaka, H.: Minimal models and abundance for positive characteristic log surfaces. Nagoya Math. J. 216, 1–70 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ye, Q.: On Gorenstein log del Pezzo surfaces. Jpn. J. Math. 28, 87–136 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceNiigata UniversityNishi-kuJapan

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