On combinatorics of plus-one generated line arrangements
Artur Bromboszcz
TL;DR
The paper addresses the combinatorial structure of plus-one generated line arrangements by developing a Poincaré-type polynomial that machinery decodes plus-one generatedness and by deriving sharp combinatorial constraints linking exponents to intersection data. It introduces a purely combinatorial non-plus-one criterion and demonstrates its effectiveness, while also providing constructive techniques via deletion from classical reflection arrangements. Concrete findings include minimal plus-one generated arrangements arising from the Klein, Wiman, and dual Hesse families, and a complete identification of nine minimal plus-one generated sporadic simplicial arrangements among those with at most 27 lines. Overall, the work deepens the connection between the combinatorics of line intersections and the homological structure of their Jacobian ideals, offering practical tools for recognizing and producing plus-one generated examples.
Abstract
In this note we focus on combinatorial aspects of plus-one generated line arrangements. We provide combinatorial constraints on such arrangements and we construct a polynomial that decodes the plus-one generated property. We present new examples of plus-one generated arrangements constructed by using classical Klein and Wiman reflection arrangements, and we detect, among all known sporadic simplicial arrangements up to $27$ lines, exactly $9$ arrangements that are minimal plus-one generated.