The Characteristic Polynomial of a Multiarrangement

Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arragnement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates ‘global’ data to ‘local’ data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free. 0. Introduction Let V be a vector space of dimension l over a field K and S = S(V ) be the symmetric algebra. We can choose coordinates for V ∗ such that S ∼= K[x1, . . . , xl]. Put ∂xi := ∂/∂xi. A hyperplane is a codimension one linear space in V . A multiarrangement is a finite collection of hyperplanes denoted by A together with a multiplicity function m : A → Z>0. Let (A,m) denote a multiarrangement. When m(H) = 1 for all H ∈ A, we identify (A,m) with the arrangement A. For p ≥ 1 the S-module Der(S) is the set of all alternating p-linear functions θ : S → S such that θ is a K-derivation in each variable. For p = 0 we put Der(S) = S. For each H ∈ A we choose a defining form αH . Put Q̃ = ∏ H∈A α m(H) H . Define the derivation modules of (A,m) as D(A,m) = {θ ∈ Der(S)|θ(αH , f2, . . . , fp) ∈ α m(H) H S for all H ∈ A and f2, . . . , fp ∈ S}. If D(A,m) is a free S-module we say that a multiarrangement (A,m) is free. One of the most fundamental invariants of an arrangement of hyperplanes is its characteristic polynomial. The focus of this paper is to generalize the characteristic polynomial to multiarrangements of hyperplanes and apply this polynomial to the problem of freeness of the module of derivations. In [17] Ziegler initiated the study Date: February 2, 2008. The first author is been supported by 21st Century COE Program “Mathematics of Nonlinear Structures via Singularities” Hokkaido University. The second author has been supported in part by Japan Society for the Promotion of Science. The third author has been supported by NSF grant # 0600893 and the NSF Japan program. 1 2 TAKURO ABE, HIROAKI TERAO, AND MAX WAKEFIELD of derivations of multiarrangements. Later in [14] and [15] Yoshinaga found that the derivation modules of multiarrangements are important for the study of free arrangements. It is known that any multiarrangement is free when l = 2 (see [12] and [17]). Other examples of free multiarrangements include the restricted multiarrangements of a free arrangement (see [17]) and the Coxeter arrangements with a constant multiplicity (see [11] and [13]). On the other hand, very few examples of non-free multiarrangements have been known. One purpose of this paper is to introduce a useful criterion for a multiarrangement to be non-free. In order to define the characteristic polynomial of a multiarrangement (A,m) we make use of the S-modules D(A,m). Since each D(A,m) is Z≥0-graded by polynomial degree, we may define a function ψ(A,m; t, q) = l