On line and pseudoline configurations and ball-quotients

TL;DR

This paper investigates the existence of certain line configurations in the projective plane whose associated Kummer covers are ball-quotients, establishing non-existence results for specific orders and configurations.

Contribution

It proves non-existence of real line configurations with ball-quotient Kummer covers of orders 3^{d-1} and 4^{d-1}, and identifies a unique configuration for order 5^{d-1}; also, it rules out certain topological configurations for small n.

Findings

No real configurations with ball-quotient Kummer covers of order 3^{d-1} for d≥4.

No real configurations with ball-quotient Kummer covers of order 4^{d-1} for d≥4.

Existence of only one configuration with order 5^{d-1}.

Abstract

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.