Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T03:20:56.135Z Has data issue: false hasContentIssue false
  • English
  • Français

Koszul Complexes and Pole Order Filtrations

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

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, ([email protected])
Gabriel Sticlaru
Affiliation:
Faculty of Mathematics and Informatics, Ovidius University, Boulevard Mamaia 124, 900527 Constanta, Romania, ([email protected])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bott, R., Homogeneous vector bundles, Annals Math. 66 (1957), 203–248.CrossRefGoogle Scholar
2.Budur, N., Dimca, A. and Saito, M., First Milnor cohomology of hyperplane arrangements, Contemp. Math. 538 (2011), 279–292.CrossRefGoogle Scholar
3.Calderon-Moreno, F. J. and Narvaez-Macarro, L., The module Df s for locally quasi-homogeneous free divisors, Compositio Math. 134 (2002), 59–74.CrossRefGoogle Scholar
4.Calderon-Moreno, F. J., Mond, D., Narvaez-Macarro, L. and Castro-Jimenez, F. J., Logarithmic cohomology of the complement of a plane curve, Comment. Math. Helv. 77 (2002), 24–38.CrossRefGoogle Scholar
5.Cheltsov, I., Factorial threefold hypersurfaces, J. Alg. Geom. 19 (2010), 781–791.CrossRefGoogle Scholar
6.Choudary, A. D. R. and Dlmca, A., Koszul complexes and hypersurface singularities, Proc. Am. Math. Soc. 121 (1994), 1009–1016.CrossRefGoogle Scholar
7.Deligne, P. and Dimca, A., Filtrations de Hodge et par l'ordre du pôle pour les hypersurfaces singulières, Annales Scient. Éc. Norm. Sup. 23 (1990), 645–656.CrossRefGoogle Scholar
8.Denham, G., Schenck, H., Schulze, M., Walther, U. and Wakefield, M., Local cohomology of logarithmic forms, Annales Inst. Fourier 63 (2013), 1177–1203.CrossRefGoogle Scholar
9.Dimca, A., On the Milnor fibrations of weighted homogeneous polynomials, Compositio Math. 76 (1990), 19–47.Google Scholar
10.Dimca, A., Differential forms and hypersurface singularities, in Singularity theory and its applications, part I, geometric aspects of singularities (ed. Mond, D. and Montaldi, J.), Springer Lecture Notes, Volume 1462, pp. 122–153 (Springer, 1991).CrossRefGoogle Scholar
11.Dimca, A., Singularities and topology of hypersurfaces (Springer, 1992).CrossRefGoogle Scholar
12.Dimca, A., Sheaves in topology (Springer, 2004).CrossRefGoogle Scholar
13.Dimca, A., Syzygies of Jacobian ideals and defects of linear systems, Bull. Math. Soc. Sci. Math. Roumanie 56(104) (2013), 191–203.Google Scholar
14.Dimca, A. and Saito, M., On the cohomology of a general fiber of a polynomial map, Compositio Math. 85 (1993), 299–309.Google Scholar
15.Dimca, A. and Saito, M., A generalization of Griffiths' theorem on rational integrals, Duke Math. J. 135 (2006), 303–326.CrossRefGoogle Scholar
16.Dimca, A. and Sticlaru, G., Chebyshev curves, free resolutions and rational curve arrangements, Math. Proc. Camb. Phil. Soc. 153 (2012), 385–397.CrossRefGoogle Scholar
17.Dimca, A. and Sticlaru, G., On the syzygies and Alexander polynomials of nodal hypersurfaces, Math. Nachr. 285 (2012), 2120–2128.CrossRefGoogle Scholar
18.Dimca, A., Saito, M. and Wotzlaw, L., A generalization of Griffiths' theorem on rational integrals, II, Michigan Math. J. 58 (2009), 603–625.CrossRefGoogle Scholar
19.Eisenbud, D., The geometry of syzygies: a second course in algebraic geometry and commutative algebra, Graduate Texts in Mathematics, Volume 229 (Springer, 2005).Google Scholar
20.Eisenbud, D. and Ulrich, B., The regularity of the conductor, in A celebration of algebraic geometry (ed. Hassett, B.et al.), Clay Mathematics Proceedings, Volume 18, pp. 267–280 (American Mathematical Society, Providence, RI, 2013).Google Scholar
21.Eisenbud, D., Green, M. and Harris, J., Cayley-Bacharach theorems and conjectures, Bull. Am. Math. Soc. 33 (1996), 295–324.CrossRefGoogle Scholar
22.Greuel, G.-M., Der Gauss–Manin–Zusammenhang isolierter Singularitäten von vollstandigen Durchschnitten, Math. Annalen 214 (1975), 235–266.CrossRefGoogle Scholar
23.Griffiths, Ph., On the period of certain rational integrals, I, II, Annals Math. 90 (1969), 460–541.CrossRefGoogle Scholar
24.Kloosterman, R., Cuspidal plane curves, syzygies and a bound on the MW-rank, J. Alg. 375 (2013), 216–234.CrossRefGoogle Scholar
25.Peters, C. and Steenbrink, J., Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Volume 52 (Springer, 2008).Google Scholar
26.Saito, K., On a generalization of de Rham lemma, Annales Inst. Fourier 26 (1976), 165–170.CrossRefGoogle Scholar
27.Saito, M., On b-function, spectrum and rational singularity, Math. Annalen 295 (1993), 51–74.CrossRefGoogle Scholar