Free line arrangements with low maximal multiplicity

TL;DR

This paper investigates the structure of free line arrangements in the complex projective plane, focusing on cases with low maximal multiplicity and classifying arrangements with small degrees.

Contribution

It characterizes free arrangements with low maximal multiplicity and describes how they can be obtained by adding or removing lines, especially for degrees up to 14.

Findings

Only two free arrangements with degree ≤14 have d₁=m+2

Descriptions of arrangements with d₁ ≤ m and d₁ = m+1

Deeper understanding of free line arrangements' geometry

Abstract

Let $\A$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\leq d_2$. Let $m$ be the maximal multiplicity of points in $\A$. In this note, we describe first the simple cases $d_1 \leq m$. Then we study the case $d_1=m+1$, and describe which line arrangements can occur by deleting or adding a line to $\A$. When $d \leq 14$, there are only two free arrangements with $d_1=m+2$, namely one with degree $13$ and the other with degree $14$. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general.