Matroid schemes and geometric posets

Abstract

The intersection data of a hyperplane arrangement is described by a geometric lattice, or equivalently a simple matroid. There is a rich interplay between this combinatorial structure and the topology of the arrangement complement. In this paper, we characterize the combinatorial structure underlying an abelian arrangement (such as a toric or elliptic arrangement) by defining a class of geometric posets and a generalization of matroids called matroid schemes. The intersection data of an abelian arrangement is encoded in a geometric poset, and we prove that a geometric poset is equivalent to a simple matroid scheme. We lay foundations for the theory of matroid schemes, discussing rank, flats, and independence. We also extend the definition of the Tutte polynomial to this setting and prove that it satisfies a deletion-contraction recurrence.