S. Yuzvinsky, FREE AND LOCALLY FREE ARRANGEMENTS WITH A GIVEN INTERSECTION LATTICE, Proceedings of the American Mathematical Society, 118(3), 1993, pp. 745-752
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.