A Description of the Resonance Variety of a Line Combinatorics via Combinatorial Pencils

In this paper we study the resonance variety of a line combinatorics. We introduce the concept of combinatorial pencil, which characterizes the components of this variety and their dimensions. The main theorem in this paper states that there is a correspondence between components of the resonance variety and combinatorial pencils. As a consequence, we conclude that the depth of a component of the resonance variety is determined by its dimension; and that there are no embedded components. This result is useful to study the isomorphisms between fundamental groups of the complements of line arrangements with the same combinatorial type. The definition of combinatorial pencil generalizes the idea of net given by Yuzvinsky and others.