Koszul complexes and pole order filtrations for projective hypersurfaces

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• Alex Dimca (Nice)
• Wednesday 09 November 2011, 14:15-15:15
• MR13, CMS.

If you have a question about this talk, please contact Burt Totaro.

I’ll discuss 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, which may be used on one hand to determine the filtration P in many cases 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 F^{2H3(U) is not equal to P}2H3(U) as soon as the degree of D is at least 4.

This talk is part of the Algebraic Geometry Seminar series.

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