Summary
Given a regular imbedding i: X → Y of codimension d, a k-dimensional variety V, and a morphism f: V → Y, an intersection product X · V is constructed in A k−d (W), W = f −1 (X). Although the case of primary interest is when f is a closed imbedding, so W = X ∩ V, there is significant benefit in allowing general morphisms f. Let g: W → X 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:
where s: W → g* N X Y is the zero-section, and s* is the Gysin map constructed in Chapter 3. Alternatively X · V is the (k − d)-dimensional component of the class
where s (W, V) is the Segre class of W in V.
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© 1984 Springer-Verlag Berlin Heidelberg
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Fulton, W. (1984). Intersection Products. In: Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02421-8_7
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DOI: https://doi.org/10.1007/978-3-662-02421-8_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-02423-2
Online ISBN: 978-3-662-02421-8
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