A Conjecture on Poincaré-Betti Series of Modules of Differential Operators on a Generic Hyperplane Arrangement

The module of derivations D(1)(A) of a hyperplane arrangement A ∈ C (henceforth called an n-arrangement) is an interesting and much studied object [10, 9, 8]. In particular, the question whether this module is free, for various classes of arrangements, has received great attention. On the other hand, the module of higher differential operators D(m)(A) received their first incisive treatment in the PhD thesis of Pär Holm [6]. The deepest result in that work concerns so-called generic arrangements, which are arrangements where every intersection of s ≤ n hyperplanes in A have the expected codimension s. Holm gave a concrete generating set for D(m)(A), proved an extension of Saito’s determinental criteria for freeness of derivations, and used these results to tackle the question of higher order freeness for generic arrangements, i.e. the question when D(m)(A) is a free module (in which case we say that A is m-free). In brief, he showed that