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

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 generalize the concept of topological configurations to a more general one (in a least possible way) such that every combinatorial configuration is realizable in this way.

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. We also generalize well-known tools from pseudoline arrangements such as sweeps and wiring diagrams. We introduce monotone quasiline arrangements as a subfamily of quasiline arrangements that can be represented with generalized wiring diagrams. We show that every incidence structure (and therefore also every combinatorial configuration) can be realized as a monotone quasiline arrangement in the real projective plane.

A quasiline arrangement with selected vertices belonging to an incidence structure can be viewed as a map on a closed surface. Such a map can be used to distinguish between two “distinct” realizations of an incidence structure as a quasiline arrangement.