An algebraic C2–equivariant Bézout theorem

Costenoble, Steven R; Hudson, Thomas; Tilson, Sean

doi:10.2140/agt.2024.24.2331

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An algebraic $C_2$–equivariant Bézout theorem

Steven R Costenoble, Thomas Hudson and Sean Tilson

Algebraic & Geometric Topology 24 (2024) 2331–2350

DOI: 10.2140/agt.2024.24.2331

Abstract

One interpretation of Bézout’s theorem, nonequivariantly, is as a calculation of the Euler class of a sum of line bundles over complex projective space, expressing it in terms of the rank of the bundle and its degree. We generalize this calculation to the $C_{2}$ –equivariant context, using the calculation of the cohomology of $C_{2}$ –complex projective spaces from an earlier paper, which used ordinary $C_{2}$ –cohomology with Burnside ring coefficients and an extended grading necessary to define the Euler class. We express the Euler class in terms of the equivariant rank of the bundle and the degrees of the bundle and its fixed subbundles. We do similar calculations using constant $ℤ$ coefficients and Borel cohomology and compare the results.

Keywords

equivariant cohomology, equivariant characteristic classes, projective space, Bézout's theorem

Mathematical Subject Classification

Primary: 55N91

Secondary: 14N10, 14N15, 55R40, 55R91

References

Publication

Received: 9 November 2022

Revised: 21 March 2023

Accepted: 21 April 2023

Published: 16 July 2024

Authors

Steven R Costenoble
Department of Mathematics Hofstra University Hempstead, NY United States

Thomas Hudson
College of Transdisciplinary Studies DGIST Daegu South Korea

Sean Tilson
Hörstel Germany

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Volume 24, issue 4 (2024)