Real and complex supersolvable line arrangements in the projective plane

We study supersolvable line arrangements in ${\mathbb{P}}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil) complex line arrangement cannot have more than 4 modular points, and if all of the crossing points of a complex line arrangement have multiplicity 3 or 4, then the arrangement must have 0 modular points (i.e., it cannot be supersolvable). This provides at least a little evidence for our conjecture that every nontrivial complex supersolvable line arrangement has at least one point of multiplicity 2, which in turn is a step toward the much stronger conjecture of Anzis and Toh\v{a}neanu that every nontrivial complex supersolvable line arrangement with $s$ lines has at least $s\mathrm{/}2$ points of multiplicity 2.