On line and pseudoline configurations and ball-quotients

Abstract

In this note we show that there are no real configurations of $d\ge 4$ 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 $d\ge 4$ 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.

7 pages, one figure. This is the final version, incorporating the suggestions of the referee, to appear in ARS Mathematica Contemporanea

References in corpus (1)

• Curve configurations in the projective plane and their characteristic numbers

Cited by in corpus (1)

• Hirzebruch-type inequalities viewed as tools in combinatorics