Hyperplane Arrangements and Their Combinatorics

Abstract

In this chapter we collect the basic definitions and results involving the intersection lattice of a hyperplane arrangement. Then we explain the key induction technique called deletion-restriction and apply it to deduce the main properties of the characteristic polynomial and of the Poincaré polynomial of an arrangement. These polynomials enter into Zaslavsky’s Theorem expressing the number of regions (resp. bounded regions) of the complement of a real arrangement. In this chapter we also introduce several important classes of hyperplane arrangements: the supersolvable arrangements, the graphic arrangements and the reflection arrangements.