The first two obstructions to the freeness of arrangements

Abstract

In his previous paper the author characterized free arrangements by the vanishing of cohomology modules of a certain sheaf of graded modules over a polynomial ring. Thus the homogeneous components of these cohomology modules can be viewed as obstructions to the freeness of an arrangement. In this paper the first two obstructions are studied in detail. In particular the component of degree zero of the first nontrivial cohomology module has a close relation to formal arrangements and to the operation of truncation. This enables us to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement.