Divisionally free arrangements of hyperplanes

Abstract

Let (A, A ′ , A H ) be the triple of hyperplane arrangements. We show that the freeness of A H and the division of χ(A; t) by χ(A H ; t) imply the freeness of A. This "division theorem" improves the famous addition-deletion theorem, and it has several applications, which include a definition of "divisionally free arrangements". It is a strictly larger class of free arrangements than the classical important class of inductively free arrangements. Also, whether an arrangement is divisionally free or not is determined by the combinatorics. Moreover, we show that a lot of recursively free arrangements, to which almost all known free arrangements are belonging, are divisionally free. Main resultsLet V be an ℓ-dimensional vector space over an arbitrary field K with ℓ ≥ 1, S = Sym(V * ) = K[x 1 , . . . , x ℓ ] its coordinate ring and Der S := ⊕ ℓ i=1 S∂ x i the module of K-linear S-derivations. A hyperplane arrangement A is a finite set of hyperplanes in V . We say that A is central if every hyperplane is linear. In this article every arrangement is central unless otherwise specified. In the central cases, we fix a linear form α H ∈ V * such that ker(α H ) = H for each H ∈ A. An ℓ-arrangement is an arrangement in an ℓ-dimensional vector space. Let L(A) := {∩ H∈B H | B ⊂ A} be an intersection lattice. L(A) has a partial order by reverse inclusion, which equips L(A) with a poset structure. For X ∈ L(A), define the localization A X of A at X by A X := {H ∈ A | H ⊃ X}, which is a subarrangement of A. Also, the restriction A X of A onto X is defined by A X := {H ∩ X | H ∈ A \ A X }, which is an *

Heavy hyperplanes in multiarrangements and their freeness

“…Hence (A, m) is free with exponents (7,9,16) by Theorem 1.2 (1). In fact, the multiarrangement (A, m + kδ H ) is free when −7 ≤ k by Theorem 1.2 (2).…”

“…Terao's conjecture asserting the dependence of the freeness only on the combinatorics is the longstanding open problem in this area. The recent approach to this problem, which gives a partial answer, is based on multiarrangements due to Yoshinaga's criterion in [18], [19], [8] and [1].…”

Solomon–Terao algebra of hyperplane arrangements

“…Theorem 2.6 (Division theorem, [1], Theorem 1.1) Let H ∈ A. If A H is free and π(A H ; t) | π(A; t), then A is free.…”

Multiple addition, deletion and restriction theorems for hyperplane arrangements

“…Theorem 1.1 ([6]) Let H ∈ A, A ′ := A \ {H} and A ′′ := A H := {H ∩ L | L ∈ A ′ }. Then two of the following three imply the third: (1) A is free with exp(A) = (d 1 , . .…”

“…There are a lot of variants of Theorem 1.1. For example, the division theorem in [1] is one of them. Among them, recently, the multiple addition theorem (MAT) is introduced in [2] to prove the freeness of ideal subarrangements.…”