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- Title
Koszul Complexes and Pole Order Filtrations.
- Authors
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 = ℙn \ 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 ℙ3, we show that F2H3(U) ≠ P2H3(U) as soon as the degree of D is at least 4.
- Publication
Proceedings of the Edinburgh Mathematical Society, 2015, Vol 58, Issue 2, p333
- ISSN
0013-0915
- Publication type
Article
- DOI
10.1017/S0013091514000182