On the geometry of real or complex supersolvable line arrangements

TL;DR

This paper proves the Dirac-Motzkin conjecture for supersolvable line arrangements, simplifies its proof, and explores connections with the slope problem and algebraic geometry over various fields.

Contribution

It provides a simplified proof of the Dirac-Motzkin conjecture for supersolvable arrangements and establishes new links with the slope problem and algebraic invariants.

Findings

The conjecture holds for all n in supersolvable arrangements.

Equivalence between the slope problem and modular point multiplicity in arrangements.

Connections between simple points and the Jacobian scheme degree.

Abstract

Given a rank 3 real arrangement $\mathcal A$ of $n$ lines in the projective plane, the Dirac-Motzkin conjecture (proved by Green and Tao in 2013) states that for $n$ sufficiently large, the number of simple intersection points of $\mathcal A$ is greater than or equal to $n/2$. With a much simpler proof we show that if $\mathcal A$ is supersolvable, then the conjecture is true for any $n$ (a small improvement of original conjecture). The Slope problem (proved by Ungar in 1982) states that $n$ non-collinear points in the real plane determine at least $n-1$ slopes; we show that this is equivalent to providing a lower bound on the multiplicity of a modular point in any (real) supersolvable arrangement. In the second part we find connections between the number of simple points of a supersolvable line arrangement, over any field of characteristic 0, and the degree of the reduced Jacobian scheme of the arrangement. Over the complex numbers even though the Sylvester-Gallai theorem fails to be true, we conjecture that the supersolvable version of the Dirac-Motzkin conjecture is true.