Free line arrangements with low maximal multiplicity

Let $\text{A}$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\le d_2$. Let $m$ be the maximal multiplicity of points in $\text{A}$. In this note, we describe first the simple cases $d_1\le m$. Then we study the case $d_1=m+1$, and describe which line arrangements can occur by deleting or adding a line to $\text{A}$. When $d\le 14$, 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.