The Freeness and Minimal Free Resolutions of Modules of Differential Operators of a Generic Hyperplane Arrangement

TL;DR

This paper proves Holm's conjecture on the freeness of modules of differential operators for generic hyperplane arrangements and constructs minimal free resolutions in cases where the modules are not free.

Contribution

It establishes conditions for the freeness of D(m)(A) and generalizes existing resolutions for modules of differential operators and jet modules.

Findings

D(m)(A) is free when n ≥ 3, r > n, m > r - n + 1.

Constructs minimal free resolutions for D(m)(A) in non-free cases.

Generalizes Yuzvinsky's and Rose-Terao's constructions for higher orders.

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 \geq 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 \geq 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.