Matroid polytopes, nested sets and Bergman fans

TL;DR

This paper explores the geometric structure of Bergman fans of matroids, their relation to nested set complexes, and the connection between different compactifications of hyperplane arrangement complements.

Contribution

It provides a geometric construction of the Bergman fan, clarifies its relationship with nested set complexes, and compares two types of compactifications in hyperplane arrangements.

Findings

Bergman fan can be constructed geometrically from matroid polytopes.

The nested set complex triangulates the Bergman complex, with equality under specific conditions.

The paper highlights subtle differences between De Concini-Procesi and tropical compactifications.

Abstract

The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De Concini-Procesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences.