The characteristic polynomial of a multiarrangement

Terao

Yoshinaga

2008

Proc. Amer. Math. Soc.

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A central arrangement A of hyperplanes in an ℓ-dimensional vector space V is said to be totally free if a multiarrangement (A, m) is free for any multiplicity m : A → Z >0 . It has been known that A is totally free whenever ℓ ≤ 2. In this article, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones.

Section: Corollary 13 Whether An Arrangement a Is Totally Free Or Nomentioning

confidence: 99%

Totally free arrangements of hyperplanes

Terao

Yoshinaga

2008

Proc. Amer. Math. Soc.

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“…Recently, some results were developed in [4] and [5] to study D(A, m) for general multiarrangements. Also, some results concerning free multiplicities on Coxeter arrangements have been found, e.g., see [3], [6] and [26].…”

Section: Introductionmentioning

confidence: 99%

Signed-eliminable graphs and free multiplicities on the braid arrangement

Journal of the London Mathematical Society

Nuida²,

Numata³

2009

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Abstract. We define specific multiplicities on the braid arrangement by using signed graphs. To consider their freeness, we introduce the notion of signedeliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of certain deformations of the braid arrangement in terms of directed graphs. IntroductionLet V = V ℓ be an ℓ-dimensional vector space over a field K of characteristic zero, {x 1 , . . . , x ℓ } a basis for the dual vector space V * and S := Sym(Vis homogeneous of degree p if f i is zero or homogeneous of degree p for each i.A hyperplane arrangement A (or simply an arrangement ) is a finite collection of affine hyperplanes in V . If each hyperplane in A contains the origin, we say that A is central. In this article we assume that all arrangements are central unless otherwise specified. A multiplicity m on an arrangement A is a map m : A → Z ≥0 and a pair (A, m) is called a multiarrangement. Let |m| denote the sum of the multiplicities H∈A m(H). When m ≡ 1, (A, m) is the same as the hyperplane arrangement A and sometimes called a simple arrangement. For each hyperplane H ∈ A fix a linear form α H ∈ V * such that ker(α H ) = H. The first main object in this article is the logarithmic derivation module D(A, m) of (A, m) defined byis free, then there exists a homogeneous free basis {θ 1 , . . . , θ ℓ } for D(A, m). Then we define the exponents of a free multiarrangement (A, m) by exp(A, m) := (deg(θ 1 ), . . . , deg(θ ℓ )). The exponents are independent of a choice of a basis. When m ≡ 1, the logarithmic derivation module and exponents are denoted by D(A) and exp(A). When we fix a simple arrangement A, we say that a multiplicity m on A is free (resp. non-free) if a multiarrangement (A, m) is free (resp. non-free).A fundamental object of study in hyperplane arrangements is the arrangement of all reflecting hyperplanes of a Coxeter group, called a Coxeter arrangement. [26]. In this article we generalize the study of free multiplicities on the braid arrangement.A braid arrangement A ℓ , or the Coxeter arrangement of type A ℓ is defined as. By using the primitive derivation introduced in [16], free multiplicities on Coxeter arrangements are studied by , Terao [21], Yoshinaga [23], and the first author and Yoshinaga [6]. Combining these results, we have a characterization of the freeness of quasi-constant multiplicities m on a Coxeter arrangement, i.e., multiplicities such that max H,H ′ ∈A |m(H) − m(H ′ )| ≤ 1. However, it is known that if max H,H ′ ∈A |m(H) − m(H ′ )| = 2 then the same method using the primitive derivation does not work. Also, to determine explicitly which multiplicity makes (A, m) free is a difficult problem. Our aim is to consider these multiplicities on the braid arrangement and classify their freeness completely. In fact...

“…In general, this definition does not work as well as it does for simple arrangements. However, if we focus on a locally heavy hyperplane, then this simply generalized Ziegler multiplicity coincides with the Euler multiplicity defined in [4]. Hence this restriction can be useful, as the following main result shows: Theorem 1.2 enables us to determine the freeness of an arbitrary unbalanced multiarrangement.…”

Section: Introductionmentioning

confidence: 99%

“…It was first defined for simple arrangements combinatorially. We only use the following algebraic characterization, which is shown in [13] for simple arrangements and in [4] for multiarrangements. For a multiarrangement (A, m), we define a function in x and t.…”

Section: Preliminariesmentioning

confidence: 99%

Heavy hyperplanes in multiarrangements and their freeness

Kühne

2017

J Algebr Comb

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Only few categories of free arrangements are known in which Terao's conjecture holds. One of such categories consists of 3-arrangements with unbalanced Ziegler restrictions. In this paper, we generalize this result to arbitrary dimensional arrangements in terms of flags by introducing unbalanced multiarrangements. For that purpose, we generalize several freeness criterions for simple arrangements, including Yoshinaga's freeness criterion, to unbalanced multiarrangements.