The freeness and minimal free resolutions of modules of differential operators of a generic hyperplane arrangement

Abstract

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

High order free hyperplane arrangements in 3-dimensional vector spaces

“…Let A be a generic ℓ-arrangement. Then A is m-free if and only if m ≥ n − ℓ + 1 [6]. The "if" part of this result coincides with Theorem 4.3 when ℓ = 3.…”

“…Holm [2] asked whether all arrangements are m-free for m large enough. It is shown that generic ℓarrangements (i.e., every ℓ hyperplanes of A intersect only at the origin) are m-free if and only if m ≥ n − ℓ + 1 [2,6]. This means that the Holm's question is true for generic arrangements.…”

A Characterization of High Order Freeness for Product Arrangements and Answers to Holm’s Questions

“…We say that A is generic if |A | > ℓ ≥ 3, and if every ℓ hyperplanes of A intersect only at the origin. For a generic arrangement A , it is shown in [7] that A is m-free if and only if m ≥ |A | − ℓ + 1. On the other hand, the behavior of m-freeness has not been well analyzed yet when m ≥ 2.…”