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

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 (Holmʼs conjecture). Combining this with some results by Holm, we see that D(m)(A) is free unless n⩾3, r>n, m