Free line arrangements with low maximal multiplicity

Alexandru Dimca; Lukas KÃ¼hne; Lukas Kühne; Piotr Pokora

doi:10.1007/s10801-026-01533-8

paper

Free line arrangements with low maximal multiplicity

arXiv:2505.01733 · doi:10.1007/s10801-026-01533-8

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.

18 pages, to appear in Journal of Algebraic Combinatorics