Combinatorial polar ordering and recursively orderable arrangements

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 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.