Hirzebruch-type inequalities and plane curve configurations

Abstract

In this paper, we come back to a problem proposed by F. Hirzebruch in the 1980s, 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 [Formula: see text]-configurations of curves in 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 [Formula: see text]-configurations. Finally, we show that the so-called characteristic numbers (or [Formula: see text] numbers) for [Formula: see text]-configurations are bounded from above by [Formula: see text]. At the end of the paper we give some examples of surfaces constructed via Kummer extensions branched along conic configurations.