Taiwanese Journal of Mathematics

Boundedness of Log Canonical Surface Generalized Polarized Pairs

Stefano Filipazzi

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In this paper, we study the behavior of the sets of volumes of the form $\operatorname{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index, and $B$ has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which $K_X+B+M$ is ample form a log bounded family. These generalize results due to Alexeev [1].

Article information

Taiwanese J. Math., Volume 22, Number 4 (2018), 813-850.

Received: 29 August 2017
Revised: 8 October 2017
Accepted: 19 December 2017
First available in Project Euclid: 4 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E30: Minimal model program (Mori theory, extremal rays) 14J10: Families, moduli, classification: algebraic theory 14J29: Surfaces of general type

generalized polarized pair volume boundedness DCC


Filipazzi, Stefano. Boundedness of Log Canonical Surface Generalized Polarized Pairs. Taiwanese J. Math. 22 (2018), no. 4, 813--850. doi:10.11650/tjm/171204. https://projecteuclid.org/euclid.twjm/1515034830

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  • V. Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810.
  • A. Beauville, Complex Algebraic Surfaces, London Mathematical Society Student Texts 34, Cambridge University Press, Cambridge, 1996.
  • C. Birkar, Anti-pluricanonical systems on Fano varieties, (2016). arXiv:1603.05765.
  • ––––, Singularities of linear systems and boundedness of Fano varieties, (2016). arXiv:1609.05543.
  • C. Birkar and D.-Q. Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Publ. Math. Inst. Hautes 'Etudes Sci. 123 (2016), 283–331.
  • S. Boucksom, T. de Fernex, C. Favre and S. Urbinati, Valuation spaces and multiplier ideals on singular varieties, in: Recent Advances in Algebraic Geometry, 29–51, London Math. Soc. Lecture Note Ser. 417, Cambridge University Press, Cambridge, 2015.
  • O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339–371.
  • ––––, Effective basepoint-free theorem for semi-log canonical surfaces, Publ. Res. Inst. Math. Sci. 53 (2017), no. 3, 349–370.
  • O. Fujino and S. Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188.
  • C. D. Hacon and S. Kovács, Classification of Higher Dimensional Algebraic Varieties, Oberwolfach Seminars 41, Birkäuser, Basel, 2010.
  • C. D. Hacon, J. M\textsuperscriptcKernan and C. Xu, On the birational automorphisms of varieites of general type, Ann. of Math. (2) 177 (2013), no. 3, 1077–1111.
  • ––––, Boundedness of varieties of log general type, (2017). arXiv:1606.07715.
  • R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
  • J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press, Cambridge, 1998.
  • R. Lazarsfeld, Positivity in Algebraic Geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 48, Springer-Verlag, Berlin, 2004.
  • S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), no. 3, 551–587.