Free line arrangements with low maximal multiplicity

TL;DR

This work analyzes freeness of line arrangements in ${\mathbb{P}}^2$ by relating exponents $(d_1,d_2)$ to the maximal multiplicity $m$ of intersection points, introducing $\varepsilon=d_1-m$. It develops a complete classification for $\varepsilon=1$ and provides general degree bounds and type constraints for $\varepsilon\ge2$, including explicit small-degree examples. Notably, it constructs two free arrangements with $d_1=m+2$ for $d=13$ and $14$, denoted $\mathcal{A}$ and $\mathcal{C}$, and proves both are divisionally free and satisfy Terao's conjecture, illustrating nuanced behavior under line deletions/additions. The paper also presents detailed geometric constructions (A13 and C14) and employs tools such as DIS sequences and plus-one generation to illuminate the structure of free line arrangements beyond purely combinatorial criteria.

Abstract

Let $\mathcal{A}$ 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 $\mathcal{A}$. 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 $\mathcal{A}$. 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.

Theorems & Definitions (26)

• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Lemma 2.4
• Theorem 2.5
• Example 2.6
• Theorem 2.7
• Example 2.8
• Remark 2.9
• Theorem 2.10