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Intersection Products

Chapter
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 2)

Summary

Given a regular imbedding i: XY of codimension d, a k-dimensional variety V, and a morphism f: VY, an intersection product X · V is constructed in A kd (W), W = f −1 (X). Although the case of primary interest is when f is a closed imbedding, so W = XV, there is significant benefit in allowing general morphisms f. Let g: WX be the induced morphism. The normal cone C W V to W in V is a closed subcone of g* N X Y, of pure dimension k. We define X · V to be the result of intersecting the k-cycle [C W V] by the zero-section of g* N X Y:
$$X\cdot V=s^{*}[C_{W}\,V]$$
where s: Wg* N X Y is the zero-section, and s* is the Gysin map constructed in Chapter 3. Alternatively X · V is the (kd)-dimensional component of the class
$$c\left ( g^{*}N_{X}Y \right)\cap s\left ( W,V \right)$$
where s (W, V) is the Segre class of W in V.

Keywords

Vector Bundle Normal Bundle Chern Class Intersection Product Cartier Divisor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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