Proceedings of the Japan Academy, Series A, Mathematical Sciences

Some remarks on log surfaces

Haidong Liu

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Fujino and Tanaka established the minimal model theory for $\mathbf{Q}$-factorial log surfaces in characteristic 0 and $p$, respectively. We prove that every intermediate surface has only log terminal singularities if we run the minimal model program starting with a pair consisting of a smooth surface and a boundary $\mathbf{R}$-divisor. We further show that such a property does not hold if the initial surface is singular.

Article information

Proc. Japan Acad. Ser. A Math. Sci. Volume 93, Number 10 (2017), 115-119.

First available in Project Euclid: 30 November 2017

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Digital Object Identifier

Primary: 14E05: Rational and birational maps
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)

$\varepsilon$-log terminal minimal model program on log surfaces


Liu, Haidong. Some remarks on log surfaces. Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 10, 115--119. doi:10.3792/pjaa.93.115.

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  • V. Alexeev, Boundedness and $K^{2}$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810.
  • M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496.
  • G. Di Cerbo, On Fujita's spectrum conjecture, arXiv:1603.09315v2.
  • O. Fujino and H. Tanaka, On log surfaces, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 8, 109–114.
  • O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339–371.
  • T. Fujita, Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 685–696.
  • J. Kollár and S. Mori, Birational geometry of algebraic varieties, translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998.
  • J. Lin, Birational unboundedness of $\mathbf{Q}$-Fano threefolds, Int. Math. Res. Not. 2003, no. 6, 301–312.
  • H. Tanaka, Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. 216 (2014), 1–70.