Hirzebruch-type inequalities and plane curve configurations

TL;DR

This paper investigates Hirzebruch-type inequalities for plane curve configurations, especially $d$-configurations, showing most do not produce ball quotients and establishing bounds on characteristic numbers, with examples of surfaces from Kummer extensions.

Contribution

It extends Hirzebruch inequalities to $d$-configurations, analyzes their desingularizations, and provides bounds on characteristic numbers, advancing understanding of complex surface configurations.

Findings

Most $d$-configurations do not yield ball quotients.

Characteristic numbers are bounded above by 8/3.

Examples of surfaces from Kummer extensions are provided.

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.