Free arrangements and rhombic tilings

Abstract

Let Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions: (1) When is the associated discriminantal hyperplane arrangement free in the sense of Saito and Terao? (2) When are all of the tilings of Z by unit rhombi coherent in the sense of Billera and Sturmfels? Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is a K (π. 1) space.