Combinatorial polar orderings and recursively orderable arrangements

TL;DR

This paper explores the combinatorics of polar orderings in hyperplane arrangements, introduces recursively orderable arrangements, and provides characterizations and properties of this class.

Contribution

It defines recursively orderable arrangements within oriented matroids, characterizes them in dimension 2, and proves all supersolvable arrangements are recursively orderable.

Findings

Complete characterization of recursively orderable arrangements in dimension 2.

Every supersolvable complexified arrangement is recursively orderable.

Abstract

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal complex is particularly easy, called {\em recursively orderable} arrangements, can therefore be combinatorially defined. We initiate the study of this class, giving a complete characterization in dimension 2 and proving that every supersolvable complexified arrangement is recursively orderable.