Free line arrangements with low maximal multiplicity

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