On combinatorics of plus-one generated line arrangements

TL;DR

This paper explores the combinatorial properties of plus-one generated line arrangements, providing constraints, a decoding polynomial, and new examples, including classical and sporadic arrangements, to advance understanding of their structure.

Contribution

It introduces combinatorial constraints, a decoding polynomial, and identifies minimal plus-one generated arrangements among known sporadic examples.

Findings

Identified 9 minimal plus-one generated arrangements among known sporadic simplicial arrangements.

Constructed new examples using Klein and Wiman reflection arrangements.

Provided a polynomial that decodes the plus-one generated property.

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.