European Journal of Mathematics

, Volume 5, Issue 3, pp 622–639 | Cite as

Log surfaces of Picard rank one from four lines in the plane

  • Valery AlexeevEmail author
  • Wenfei Liu
Research Article


We derive simple formulas for the basic numerical invariants of a singular surface with Picard number one obtained by blow-ups and contractions of the four-line configuration in the plane. As an application, we establish the smallest positive volume and the smallest accumulation point of volumes of log canonical surfaces obtained in this way.


Log canonical surfaces Volume Four-line configuration 

Mathematics Subject Classification

14J29 14J26 14R05 



  1. 1.
    Alexeev, V.: Classification of log canonical surface singularities: arithmetical proof. In: Flips and Abundance for Algebraic Threefolds. Astérisque, vol. 211, pp. 47–58. Société Mathématique de France, Paris (1992)Google Scholar
  2. 2.
    Alexeev, V.: Boundedness and \(K^2\) for log surfaces. Internat. J. Math. 5(6), 779–810 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alexeev, V., Liu, W.: Open surfaces of small volume. Algebraic Geom. 6(3), 312–327 (2019).
  4. 4.
    Alexeev, V.A., Liu, W.: On accumulation points of volumes of log surfaces. Izv Math. 83 (2019).
  5. 5.
    Alexeev, V., Mori, S.: Bounding singular surfaces of general type. In: Christensen, C., et al. (eds.) Algebra, Arithmetic and Geometry with Applications, pp. 143–174. Springer, Berlin (2004)CrossRefGoogle Scholar
  6. 6.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)zbMATHGoogle Scholar
  7. 7.
    The Sage Developers: Sagemath, the Sage Mathematics Software System (Version 7.5.1) (2017).

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenPeople’s Republic of China

Personalised recommendations