A generalized Desargues configuration and the pure braid group

In this paper, a configuration with n = (₂^d) points in the plane is described. This configuration, as a matroid, is a Desargues configuration if d = 5, and the union of (₅^d) such configurations if d> 5. As an oriented matroid, it is a rank 3 truncation of the directed complete graph on d vertices. From this fact, it follows from a version of the Lefschetz-Zariski theorem implied by results of Salvetti that the fundamental group π of the complexification of its line arrangement is Artin's pure (or coloured) braid group on d strands.

In this paper we obtain, by using techniques introduced by Salvetti, a new algorithm for finding a presentation of π based on this particular configuration.