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Cohomological and motivic inclusion–exclusion

Part of: Special properties Surfaces and higher-dimensional varieties Fiber spaces and bundles

Published online by Cambridge University Press:  13 September 2024

Ronno Das Open the ORCID record for Ronno Das [Opens in a new window]  and
Sean Howe Open the ORCID record for Sean Howe [Opens in a new window]
Ronno Das
Affiliation:
Matematiska institutionen, Stockholm University, 106 91 Stockholm, Sweden [email protected]
Sean Howe
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA [email protected]
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Abstract

We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.

Keywords

homological stabilityhypersurfacesinclusion–exclusionmotivic statisticsstratified spaces

MSC classification

Primary: 55R80: Discriminantal varieties, configuration spaces
Secondary: 14J70: Hypersurfaces 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 9 , September 2024 , pp. 2228 - 2283
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Copyright
© The Author(s), 2024. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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