The e-multiplicity and addition-deletion theorems for multiarrangements

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 e-multiplicity, of a restricted multiarrangement. We compute the e-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. 0 Introduction Let A be a hyperplane arrangement, or simply an arrangement. In other words, A is a finite collection of hyperplanes in an `-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∈Am(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 {x1, . . . , x`} be a basis for V ∗. Then S := Sym(V ∗) ' K[x1, . . . , x`]. When each H ∈ A contains the origin, we say that A is central. Throughout this article, assume that every arrangement is central. Let DerK(S) denote the set of K-linear derivations from S to itself. For each H ∈ A we choose a defining form αH . Following Ziegler [Z], we define an S-module D(A,m) of a multiarrangement (A,m) by D(A,m) = {θ ∈ DerK(S) | θ(αH) ∈ α H S for all H ∈ A}. If D(A,m) is a free S-module we say that (A,m) is a free multiarrangement. When (A,m) is simple, the module coincides with the usual module D(A) of logarithmic derivations (e.g., [OT, 4.1]). Thus free multiarrangements generalize free arrangements. ∗Supported by 21st Century COE Program “Mathematics of Nonlinear Structures via Singularities” Hokkaido University. †Supported in part by Japan Society for the Promotion of Science. ‡Supported by NSF grant # 0600893 and the NSF Japan program.