A Graph Theoretical Perspective on Hyperplane Arrangement Theory

A hyperplane arrangement is called free if its module of logarithmic derivations is free. The class of subarrangements of the well-studied braid arrangement, so-called graphic hyperplane arrangements, is important for analysing new concepts and properties in hyperplane arrangement theory. A graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. We extend this result by proving that the module of logarithmic derivations of an arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices. Furthermore, based on observations from the proof of the result above, we introduce an explicit set of generators for the module of logarithmic derivations of a general graphic arrangement based on graph separators and connectivity. We obtain new insights in the derivation degree sequence in the non-free case and give bounds on the highest degree in the sequence. Moreover, this can broadly be used as a tool for future research in this area. The module of logarithmic derivations of the more general case of hypersurfaces is used in the development of novel tools for computing the likelihood correspondence of an arrangement of hypersurfaces in a projective space. We exemplify this concept through use of graphic arrangements and broaden the focus to nonlinear scenarios and their applications in statistics and physics. Another field of application for hyperplane arrangements is the classification of combinatorial objects, for example metrics. In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport and are called “Wasserstein polytopes” or “Kantorovich-Rubinstein polytopes” in the literature. Answering a question posed by Vershik, we describe the stratification of the metric cone induced by the combinatorial type of these polytopes via an associated hyperplane arrangement. Moreover, we study its relationship with the stratification by combinatorial type of injective hulls (i.e., tight spans) and, in particular, with certain types of metrics arising in phylogenetic analysis. We also compute enumerative invariants in the case of metrics on up to six points.