Combinatorial configurations, quasiline arrangements, and systems of curves on surfaces

TL;DR

This paper introduces quasiline arrangements as a new topological framework to realize all combinatorial configurations on surfaces, generalizing pseudoline arrangements and extending classical tools like sweeps and wiring diagrams.

Contribution

It generalizes pseudoline arrangements to quasiline arrangements, enabling topological realizations of all combinatorial configurations on surfaces.

Findings

Every combinatorial configuration can be realized as a quasiline arrangement in the real projective plane.

Generalization of sweep and wiring diagram tools to quasiline arrangements.

Use of surface maps to distinguish between different realizations of configurations.

Abstract

It is well known that not every combinatorial configuration admits a geometric realization with points and lines. Moreover, some of them do not even admit realizations with pseudoline arrangements, i.e., they are not topological. In this paper we provide a new topological representation by using and essentially generalizing the topological representation of oriented matroids in rank 3. These representations can also be interpreted as curve arrangements on surfaces. In particular, we generalize the notion of a pseudoline arrangement to the notion of a quasiline arrangement by relaxing the condition that two pseudolines meet exactly once and show that every combinatorial configuration can be realized as a quasiline arrangement in the real projective plane. We also generalize well-known tools from pseudoline arrangements such as sweeps or wiring diagrams. A quasiline arrangement with selected vertices belonging to the configuration can be viewed as a map on a closed surface. Such a map can be used to distinguish between two "distinct" realizations of a combinatorial configuration as a quasiline arrangement.