# Real and complex supersolvable line arrangements in the projective plane

## TL;DR

This paper investigates the structure of supersolvable line arrangements in the projective plane over reals and complex numbers, establishing bounds on modular points and multiplicities to support conjectures on their combinatorial classification.

## Contribution

It proves that nontrivial complex supersolvable arrangements have at most four modular points and that arrangements with only triple or quadruple points lack modular points, advancing classification conjectures.

## Key findings

- Complex arrangements have at most 4 modular points.
- Arrangements with only 3- or 4-multiplicity points have no modular points.
- Supports conjecture that supersolvable arrangements have many points of multiplicity 2.

## Abstract

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 $s/2$ points of multiplicity 2.