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 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.

Some remarks on free arrangements

“…1.6]. The following confirms a conjecture of Abe, [Abe18,Conj. 4.4], which resolves the containment relations among these classes of free arrangements.…”

“…In his recent papers [Abe17] and [Abe18], T. Abe shows that all free arrangements that obey Terao's Addition-Deletion Theorem 2.3 are indeed combinatorial. In [Abe18], he introduced a new class of free arrangements, so called stair-free arrangements SF (Definition 2.9). Its significance lies in the fact that Terao's Conjecture 1.1 is still valid within SF ([Abe18, Thm.…”

“…To date this is the largest known class of free arrangements with this property. This class encompasses the class of divisionally free arrangements DF (Definition 2.7) and the class of additionally free arrangements AF (Definition 2.8), [Abe18,Thm. 4.3].…”

Projective dimensions of hyperplane arrangements

“…Thus applying Theorem 1.17 to (A X , H) for all X ∈ L 2 (A H ), we can show that (A X , H) ∈ CS 3 , and A ′ ∈ IPD 4 1 , thus A ′ is NMPD along H by Proposition 5.5. Also, A H ∈ PDC 4 2 .…”

On the non-connectivity of moduli spaces of arrangements: the splitting-polygon structure

“…As mentioned in the abstract, the questions related to the combinatorial nature of some properties of a hyperplane arrangement are numerous in the literature. If some of them have been solved by the affirmative, as for the number of chambers of a real arrangement [27], the cohomology ring of the complement [19], the rank of the lower central series quotients of the fundamental group of its complement [10] or the deletion and addition-deletion theorems of free arrangements [2,1]; some others obtained a negative answer, as for the embedded topology of a complex arrangements or the fundamental group of its complements, see [21,12,5,13], (also negative for the smaller class of real complexified arrangements [3,14]), the torsion of the lower central series quotients [8] or the existence of unexpected curves [14]. Naturally, the number of problems which are still open (or conjectural) is larger; like the famous Terao's conjecture [24,20], the combinatorial nature of the characteristic varieties [15] or of the homology of the Milnor fiber [11,Problem 4.5], to name some but a few.…”