# Modular Construction of Free Hyperplane Arrangements

## TL;DR

This paper explores the construction of free hyperplane arrangements using modular joins, generalizing known results for graphic arrangements and applying to gain graphs and finite fields.

## Contribution

It generalizes Dirac's construction to simple matroids with modular joins, establishing divisional freeness for a broad class of arrangements.

## Key findings

- Arrangements from modular joins are divisionally free.
- Generalization of Dirac's construction to matroids.
- Applications to gain graphs and finite field arrangements.

## Abstract

In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac showed that a graph is chordal if and only if the graph is obtained by "gluing" complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields.