Koszul complexes and pole order filtrations

We study 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 $U={\text{PP}}^n\setminus D$, 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 ${\text{PP}}^3$, we show that $F^2{H}^3(U)\mathrm{\ne }{P}^2{H}^3(U)$ as soon as the degree of $D$ is at least 4.