Broken circuit complexes and hyperplane arrangements

TL;DR

This paper investigates algebraic properties of broken circuit complexes and hyperplane arrangements, linking combinatorial structures to algebraic resolutions, and applying these insights to graph theory and matroid theory.

Contribution

It characterizes when Stanley-Reisner ideals of broken circuit complexes have linear resolutions or are complete intersections, and connects these to properties of hyperplane arrangements and matroids.

Findings

Identifies conditions for linear resolutions and complete intersections in broken circuit complexes.

Characterizes arrangements with Orlik-Terao ideals sharing these properties.

Improves bounds on chromatic polynomial coefficients for maximal planar graphs.

Abstract

We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.