Addition-deletion theorem for free hyperplane arrangements and combinatorics

Abstract

In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this article, we prove that Terao's celebrated addition theorem for free arrangements is combinatorial. Combining other developments in these days, we can show that all the addition-deletion framework is combinatorial. As a corollary, we can define a new class of free arrangements called the additively free arrangement of hyperplanes, which can be constructed from the empty arrangement by using only the addition theorem. Then we can show that Terao's conjecture is true in this class. As an application, we prove that the freeness of all ideal-Shi arrangements is combinatorial.