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arXiv · 2505.01733

Free line arrangements with low maximal multiplicity

Alexandru Dimca·Lukas Kühne·Piotr Pokora
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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.

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Alexandru Dimca, Lukas Kühne, Piotr Pokora. 2025-05-03. Free line arrangements with low maximal multiplicity. https://doi.org/10.1007/s10801-026-01533-8

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