FREE AND LOCALLY FREE ARRANGEMENTS WITH A GIVEN INTERSECTION LATTICE

In previous papers the author characterized free arrangements of hyper planes by the vanishing of cohomology of the intersection lattice with coefficients in a certain sheaf of graded modules over a polynomial r ing. The main result of this paper is that for a locally free arrangem ent the degrees of nonzero homogeneous components of the cohomology mo dules are bounded by a number depending only on the intersection latti ce. In particular, the Hilbert coefficients of the module of derivatio ns of a locally free arrangement are combinatorial invariants. Another result of the paper asserts that the set of free arrangements is Zari ski open in the set of all arrangements with a given intersection latt ice.