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Koszul Complexes and Pole Order Filtrations

Part of: Cycles and subschemes Singularities Commutative algebra: Homological methods

Published online by Cambridge University Press:  27 October 2014

Alexandru Dimca  and
Gabriel Sticlaru
Alexandru Dimca
Affiliation:
Laboratoire de Mathématiques J. A. Dieudonné, Unité Mixte de Recherche 7351, Centre National pour la Recherche Scientifique, University of Nice Sophia Antipolis, 06100 Nice, France, (dimca@unice.fr)
Gabriel Sticlaru
Affiliation:
Faculty of Mathematics and Informatics, Ovidius University, Boulevard Mamaia 124, 900527 Constanta, Romania, (gabrielsticlaru@yahoo.com)
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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.

Keywords

projective hypersurfacessingularitiesMilnor algebrasyzygiesmixed Hodge structurepole order filtration

MSC classification

Primary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Secondary: 32S35: Mixed Hodge theory of singular varieties 13D02: Syzygies, resolutions, complexes
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 333 - 354
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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