# Triangular arrangements on the projective plane

## TL;DR

This paper investigates triangular line arrangements in the projective plane, showing their combinatorics are realized by Roots-of-Unity-Arrangements and exploring conditions for their freeness.

## Contribution

It establishes that all combinatorics of triangular arrangements are realized by Roots-of-Unity-Arrangements and analyzes their freeness conditions.

## Key findings

- All combinatorics of triangular arrangements are realized by Roots-of-Unity-Arrangements.
- Conditions for freeness of Roots-of-Unity-Arrangements are provided.
- Examples show arrangements with identical weak combinatorics can differ in freeness.

## Abstract

In this work we study line arrangements consisting in lines passing through three non-aligned points. We call them triangular arrangements. We prove that any combinatorics of a triangular arrangement is always realized by a Roots-of-Unity-Arrangement, which is a particular class of triangular arrangements. Among these Roots-of Unity-Arrangements, we provide conditions that ensure their freeness. Finally, we give two triangular arrangements having the same weak combinatorics, such that one is free but the other one is not.