Hirzebruch-type inequalities and plane curve configurations

Abstract

In this paper we come back to a problem proposed by F. Hirzebruch in the 1980's, namely whether there exists a configuration of smooth conics in the complex projective plane such that the associated desingularization of the Kummer extension is a ball quotient. We extend our considerations to the so-called $d$-configurations of curves on the projective plane and we show that in most cases for a given configuration the associated desingularization of the Kummer extension is not a ball quotient. Moreover, we provide improved versions of Hirzebruch-type inequality for $d$-configurations. Finally, we show that the so-called characteristic numbers (or $\gamma$ numbers) for $d$-configurations are bounded from above by $8/3$. At the end of the paper we give some examples of surfaces constructed via Kummer extensions branched along conic configurations.

9 pages, Final version, to appear in International Journal of Mathematics

Cited by in corpus (4)

• Conic-line arrangements in the complex projective plane
• The 21 reducible polars of Klein's quartic
• Hirzebruch-type inequalities viewed as tools in combinatorics
• Bounds for characteristic numbers of conic-line arrangements in the plane