Combinatorics of free and simplicial line arrangements

TL;DR

This paper investigates the combinatorial properties of pseudoline arrangements in the real projective plane, focusing on simplicial arrangements and those with real-rooted characteristic polynomials, deriving inequalities, classification results, and proving the Dirac Motzkin Conjecture in specific cases.

Contribution

It introduces new inequalities for the t-vectors of pseudoline arrangements and proves the Dirac Motzkin Conjecture for arrangements with characteristic polynomials splitting over the reals.

Findings

Derived inequalities involving t-vectors of arrangements.

Obtained finiteness and classification results for certain arrangements.

Proved the Dirac Motzkin Conjecture for arrangements with real-rooted characteristic polynomials.

Abstract

We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive inequalities involving the $t$-vectors of the arrangements in consideration. As application, we obtain some finiteness and classification results. Moreover, we are able to prove the Dirac Motzkin Conjecture for real pseudoline arrangements whose charateristic polynomials split over $\mathbb{R}$.