On generators of the module of logarithmic 1-forms with poles along an arrangement (Q1893957)

The author considers elements \(X\) of codimension 2 of the intersection lattice of an arrangement \(A\) of \(n\) hyperplanes in a \(d\)-dimensional linear space over an arbitrary field. Defining differential logarithmic 1-forms \(\omega_X\) for these elements, with poles along \(A\), he describes the class of arrangements for which forms \(\omega_X\) generate the whole module of the logarithmic 1-forms with poles along \(A\). This is done in terms of linear relations among the functionals defining the hyperplanes. Further on, a minimal free resolution of the module generated by \(\omega_X\) is constructed, that in particular defines the projective dimension of this module. For studying relations among \(\omega_X\), free resolutions of certain ideals of a polynomial ring, generated by products of linear forms, are constructed. The author presents certain 3- and 4-arrangements as examples, and he finally discusses two possible directions of generalization.