Koszul Complexes and Pole Order Filtrations. Issue 2 (27th October 2014)

• Record Type: Journal Article
• Authors: Dimca, Alexandru; Sticlaru, Gabriel
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 58:Issue 2(2015)
• Page Start: 333
• Page End: 354
• Publication Date: 2014-10-27
• Subjects: projective hypersurfaces, -- singularities, -- Milnor algebra, -- syzygies, -- mixed Hodge structure, -- pole order filtration; Primary 14C30, -- 13D40, -- Secondary 32S35, -- 13D02; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091514000182
• Languages: English
• ISSNs: 0013-0915

Abstract

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.