Addition-deletion theorem for free hyperplane arrangements and combinatorics

TL;DR

This paper proves that Terao's addition-deletion theorem for free hyperplane arrangements depends only on combinatorial data, establishing the conjecture for a new class of arrangements and showing the freeness of ideal-Shi arrangements.

Contribution

It demonstrates the combinatorial nature of Terao's addition theorem and introduces the class of additionally free arrangements, confirming Terao's conjecture within this class.

Findings

Terao's addition-deletion theorem is combinatorial.

Introduces additionally free arrangements constructed via addition theorem.

Shows ideal-Shi arrangements are additionally free, confirming their freeness is combinatorial.

Abstract

In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally 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 can show that every ideal-Shi arrangement is additionally free, implying that their freeness is combinatorial.