Freeness of hyperplane arrangement bundles and local homology of arrangement complements

A recent result of Salvetti and Settepanella gives, for a complexified real arrangement, an explicit description of a minimal CW decomposition as well as an explicit algebraic complex which computes local system homology. We apply their techniques to discriminantal arrangements in two dimensional complex space and calculate the boundary maps which will give local system homology groups given any choice of local system. This calculation generalizes several known results; examples are given related to Milnor fibrations, solutions of KZ equations, and the LKB representation of the braid group.

Another algebraic object associated to a hyperplane arrangement is the module of derivations. We analyze the behavior of the derivation module for an affine arrangement over an infinite field and relate its derivation module to that of its cone. In the case of an arrangement fibration, we analyze the relationship between the derivation module of the total space arrangement and those of the base and fiber arrangements. In particular, subject to certain restrictions, we establish freeness of the total space arrangement given freeness of the base and fiber arrangements.