Free line arrangements with low maximal multiplicity

2025/05/03 by Dimca, Alexandru, Kühne, Lukas, Pokora, Piotr
#13D02 #14H50 #32S25 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.01733

Abstract

Let A be a free arrangement of d lines in the complex projective plane, with exponents d₁≤ d₂. Let m be the maximal multiplicity of points in A. In this note, we describe first the simple cases d₁ ≤ m. Then we study the case d₁=m+1, and describe which line arrangements can occur by deleting or adding a line to A. When d ≤ 14, there are only two free arrangements with d₁=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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