The Module of Derivations for an Arrangement of Subspaces

Let V be a linear space of dimension over a field K. By an arrangement we shall mean a finite collection of affine subspaces of V . If all of the subspaces in an arrangement A have codimension k then we say that A is an ( , k)arrangement. If k = 1 and so A is a hyperplane arrangement then we shall say that A is an -arrangement. Let A be an arrangement and S the coordinate ring for V . For each H ∈ A let IH = V(H), the ideal of S which vanishes on H, and call it the defining ideal for H. If H is a hyperplane, then we can choose a linear functional αH ∈ S such that IH = (αH). We now introduce the main character of this paper. IfA is an arrangement then the module of A-derivations is D(A), the set of all K-linear derivations of S which map each defining ideal to itself. Equivalently, one could define D(A) to be the set of all polynomial vector fields which, at each subspace, are parallel to that subspace. [1] contains an extensive review of the properties of D(A) for hyperplane arrangements, especially for free arrangements. We shall review the situation for generic arrangements in Section 3. Recently interest has arisen in arrangements of subspaces of codimension greater than one. The goal of this paper is to examine D(A) in this case. In particular we investigate subspace arrangements consisting of elements of the intersection lattice of a generic hyperplane arrangement, where we find generators for D(A) as an S-module. In Section 2 we list several elementary properties of D(A). In Section 3 we find generators of D(A) for generic arrangements. In Section 4 we discuss