Hirzebruch–Kummer covers of algebraic surfaces

Hirzebruch–Kummer covers of algebraic surfaces

The aim of this paper is to show that using some natural curve arrangements in algebraic surfaces andHirzebruch–Kummer covers, one cannot construct new examples of ball-quotients, i.e. minimal smooth complex projectivesurfaces of general type satisfying equality in the Bogomolov–Miyaoka–Yau inequality.

___

  • [1] Barthel G, Hirzebruch F, Höfer T. Geradenkonfigurationen und algebraische Flächen. Aspects of mathematics. D4. Vieweg, Braunschweig, 1987 (in German).
  • [2] Catanese F. Kodaira fibrations and beyond: methods for moduli theory. Japan J Math 2017; 12: 91-174.
  • [3] Eterović S. Logarithmic Chern slopes of arrangements of rational sections in Hirzebruch surfaces. MS, Pontificia Universidad Católica de Chile, Santiago, Chile, 2015.
  • [4] Hirzebruch F. Arrangements of lines and algebraic surfaces. In: Artin M, Tate J, editors. Arithmetic and Geometry, Vol II, Progr Math vol. 36, Boston, MA, USA: Birkhäuser, 1983, pp. 113-140.
  • [5] Hirzebruch F. Chern numbers of algebraic surfaces. Math Ann 1984; 266: 351-356.
  • [6] Hirzebruch F. Algebraic surfaces with extreme Chern numbers (report on the thesis of Th. Höfer, Bonn 1984). Russian Math Surveys 1985; 40: 135-145.
  • [7] Hirzebruch F. Singularities of algebraic surfaces and characteristic numbers. The Lefschetz centennial conference, Part I (Mexico City, 1984) Contemp Math 1986; 58: 141-155.
  • [8] Hunt B. Coverings and ball quotients with special emphasis on the 3-dimensional case. Bonner Mathematische Schriften 1986; 174.
  • [9] Miyaoka Y. The maximal number of quotient singularities on surfaces with given numerical invariants. Math Ann 1984; 268: 159-171.
  • [10] Pardini R. Abelian covers of algebraic varieties. J Reine Angew Math 1991; 417: 191-213.
  • [11] Pokora P. Hirzebruch type inequalities and plane curve configurations. Internat J Math 2017; 28: 1750013 (11 pages).
  • [12] Roulleau X. Divisor arrangements and algebraic surfaces. Comment Math Univ St Pauli 2008; 57: 13-21.
  • [13] Roulleau X. Bounded negativity, Miyaoka-Sakai inequality and elliptic curve configurations. Int Math Res Not 2017; 2017: 2480-2496.
  • [14] Urzúa G. Arrangements of curves and algebraic surfaces. PhD, University of Michigan, Ann Arbor, MI, USA, 2008.
  • [15] Urzúa G. Arrangements of rational sections over curves and the varieties they define. Rend Lincei Mat Appl 2011; 22: 453-486.