The Euler multiplicity and addition–deletion theorems for multiarrangements

Abstract

The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements including supersolvable arrangements and the Coxeter arrangement of type A3 to construct free and non-free multiarrangements. IntroductionLet A be a hyperplane arrangement, or simply an arrangement. In other words, A is a finite collection of hyperplanes in an ℓ-dimensional vector space V over a field K. A multiarrangement, which was introduced by Ziegler in [Z], is a pair (A, m) consisting of a hyperplane arrangement A and a multiplicity m : A → Z >0 . Define |m| = H∈A m(H). A multiarrangement (A, m) such that m(H) = 1 for all H ∈ A is just a hyperplane arrangement, and is sometimes called a simple arrangement.Let {x 1 , . . . , x ℓ } be a basis forWhen each H ∈ A contains the origin, we say that A is central. Throughout this article, assume that every arrangement is central. Let Der K (S) denote the set of K-linear derivations from S to itself. For each H ∈ A we choose a defining formIf D(A, m) is a free S-module we say that (A, m) is a free multiarrangement. In his groundbreaking paper [Z], Ziegler writes ". . . the theory of multiarrangements and their freeness is not yet in a satisfactory state. In particular, we do not know any addition/deletion theorem . . . ." It is exactly the subject of this article. Namely, we generalize the addition-deletion theorems for simple arrangements [T1] to multiarrangements in this article. Let (A, m) be a nonempty multiarrangement and ℓ ≥ 2. Fix a hyperplane H 0 ∈ A and let α 0 be a defining form for H 0 . To state the addition-deletion theorems for multiarrangements we need to define the deletion (A ′ , m ′ ) and the restriction (A ′′ , m ′′ ). First, we define the deletion as follows:Next we define the restriction (A ′′ , m ′′ ). Letwhich is an arrangement on H 0 . We, however, have more than one choice to define a multiplicity m ′′ . The definition of a suitable multiplicity m ′′ is crucial. The canonical definition is probablywhich is purely combinatorial and was used in [Y1, Y2, Z] effectively. In this article, however, in order to serve our purposes, we introduce a new multiplicity m * , called the Euler multiplicity, whose definition is algebraic rather than combinatorial.For X ∈ A ′′ define A X = {H ∈ A | X ⊂ H} and m X = m | AX .

Totally free arrangements of hyperplanes

“…The third author proved in [9] and [10] that the freeness of a simple arrangement is closely related with the freeness of Ziegler's canonical restriction. Recently the first and second authors and Wakefield developed a general theory of free multiarrangements and introduced the concept of free multiplicity in [3] and [4]. Several papers including [1], [2], [5] and [11] studied the set of free multiplicities for a fixed arrangement A.…”

Heavy hyperplanes in multiarrangements and their freeness

“…is a rank 2 arrangement and we can choose a basis [5]. Then the Euler multiplicity m * on A H 0 is defined as m * (X) := pdeg θ X and (A H 0 , m * ) is called the Euler restriction of (A, m).…”

“…The determination of the freeness of a given multiarrangement is far more difficult than that of a simple arrangement. There are only two ways to validate it: Ziegler's free restriction theorem in [20], or the addition-deletion theorem in [5]. To apply the former, we need to find a free simple arrangement whose Ziegler restriction is the given one.…”

“…To apply the former, we need to find a free simple arrangement whose Ziegler restriction is the given one. To apply the latter, we need a free multiarrangement close to the given one and then increase/decrease multiplicities by applying the addition-deletion theorem in [5]. Theorem 1.2 enables us to check the freeness only by using the information of the given one under the heaviness assumption.…”

On the Extendability of Free Multiarrangements

“…H possesses a natural multiplicity Recently freeness of multiarrangements are extensively studied [3,4,12,15,16,17]. The motivation to this article is to ask whether if a free multiarrangement is obtained as a restriction of a free simple arrangement.…”