Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D((m))(A) of the nth Weyl algebra of homogeneous differential operators of order m preserving the defining ideal of A.
We prove that if n >= 3, r > n, m > r - n + 1, then D((m))(A) is free (Holm's conjecture). Combining this with some results by Holm, we see that D((m))(A) is free unless n >= 3, r > n, m < r - n + 1. In the remaining case, we construct a minimal free resolution of D((m))(A) by generalizing Yuzvinsky's construction for m = 1. In addition, we construct a minimal free resolution of the transpose of the m-jet module, which generalizes a result by Rose and Terao for m = 1. (C) 2011 Elsevier Inc. All rights reserved.