Hyperplane arrangements as models for combinatorial geometry

Positive geometries might be studied as generalizations of polytopes, and we’d like to import insights from combinatorial geometries associated with polytopes. A well-understood first case is that of hyperplane arrangements: there, the combinatorics of the arrangement controls much of the topology and arithmetic of the complement. We’ll survey relevant constructions and calculations for hyperplane arrangements, such as the characteristic polynomial, the cohomology ring, and point-counts over finite fields. Taking those as a model, one could later apply similar ideas to other positive geometries.

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Date published: Wednesday, November 1, 2023