On Generators of the Module of Logarithmic 1-Forms with Poles Along an Arrangement

For each element X of codimension two of the intersection lattice of a hyperplane arrangement we define a differential logarithmic 1-forms tax with poles along the arrangement. Then we describe the class of arrangements for which forms wX generate the whole module of the logarithmic 1-forms with poles along the arrangement. The description is done in terms of linear relations among the functionals defining the hyperplanes. We construct a minimal free resolution of the module generated by wx that in particular defines the projective dimension of this module. In order to study relations among wx we construct free resolutions of certain ideals of a polynomial ring generated by products of linear forms. We give examples and discuss possible generalizations of the results.