  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.
