Matroid schemes and geometric posets

TL;DR

This paper introduces matroid schemes and geometric posets to generalize the combinatorial structures of hyperplane arrangements, extending classical matroid theory to abelian arrangements like toric and elliptic cases.

Contribution

It defines matroid schemes and geometric posets, establishing their equivalence and extending matroid concepts and the Tutte polynomial to abelian arrangements.

Findings

Geometric posets encode intersection data of abelian arrangements.

Matroid schemes generalize simple matroids for abelian arrangements.

Tutte polynomial extended with deletion-contraction recurrence.

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.