In this note we show that there are no real configurations of $d\geq 4$ lines in the projective plane such that the associated Kummer covers of order $3^{ d-1}$ are ball-quotients and there are no configurations of $d\geq 4$ lines such that the Kummer covers of order $4^{ d-1}$ are ball-quotients. Moreover, we show that there exists only one configuration of real lines such that the associated Kummer cover of order $5^{ d-1}$ is a ball-quotient. In the second part we consider the so-called topological $(n_{ k})$-configurations and we show, using Shnurnikov's inequality, that for $n < 27$ there do not exist $(n_{ 5})$-configurations and and for $n < 41$ there do not exist $(n_{ 6})$-configurations.
Categories: math.AG math.CO