Koszul Complexes and Pole Order Filtrations. Issue 2 (27th October 2014) - Dimca, Alexandru; Sticlaru, Gabriel
Abstract: 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 = ℙ n D, with D the hypersurface defined by f = 0. The relation is expressed by some spectral sequences. These sequences may, on the one hand, in many cases be used to determine the filtration P 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 2 H 3 ( U ) ≠ P 2 H 3 ( U ) as soon as the degree of D is at least 4.