Simplicial arrangements with few double points

TL;DR

This paper investigates the structure of simplicial line arrangements with few double points, showing that the associated cubic curve must be reducible, thus supporting Grünbaum's classification under certain conditions.

Contribution

It proves that in simplicial arrangements with few double points, the cubic curve involved cannot be irreducible, advancing the understanding of their geometric structure.

Findings

The cubic curve associated with such arrangements is reducible.

Supports Grünbaum's conjectural classification under a linear double point bound.

Provides geometric proofs linking arrangement properties to algebraic curve reducibility.

Abstract

In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. It follows that Gr\"{u}nbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points.