Real and complex supersolvable line arrangements in the projective plane

Abstract

If we regard a set of s lines in \({\mathbb P}^2\) over either the reals or the complex numbers as an algebraic plane curve, then it is an open problem to classify for all s those for which the number \(t_2\) of points of multiplicity 2 satisfies \(t_2<\lfloor s/2\rfloor \). By the Sylvester–Gallai theorem, there are no nontrivial (i.e., not a pencil or a near pencil) real arrangements with \(t_2=0\), but there are complex arrangements with \(t_2=0\) and it is an open problem to classify them. In this paper, we initiate a classification of an interesting class of line arrangements called the supersovable line arrangements and give a partial classification for them over the reals or the complex numbers. In particular, we show that a complex line arrangement which is nontrivial cannot have more than 4 modular points and we completely describe those with 3 or 4 modular points.