| I'll discuss the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial f and the pole order filtration P on the cohomology of the open set U given by the projective space minus D, with D the hypersurface defined by f=0. The relation is expressed by some spectral sequences, which may be used on one hand to determine the filtration P in many cases for curves and surfaces, and on the other hand to obtain information about the syzygies involving the partial derivatives of the polynomial f. The case of a nodal hypersurface D is treated in terms of the defects of linear systems of hypersurfaces of various degrees passing through the nodes of D. When D is a nodal surface in P^3, we show that F^2H^3(U) is different from P^2H^3(U) as soon as the degree of D is at least 4. |