A greedy algorithm to compute arrangements of lines in the projective plane

TL;DR

This paper presents a greedy algorithm for optimizing line arrangements in the projective plane, successfully recovering known configurations and discovering new ones, including a 35-line simplicial arrangement, and building a database of complex arrangements.

Contribution

It introduces a novel greedy algorithm for line arrangements and applies it to find new simplicial configurations and compile a comprehensive database.

Findings

Recovered all known simplicial arrangements efficiently.

Discovered a new 35-line simplicial arrangement.

Compiled a database with arrangements up to 50 lines, including some with infinite realization spaces.

Abstract

We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.