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Excess and Residual Intersections

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

Summary

If XY is a regular imbedding, VY a subvariety, we have constructed (§ 6.1) an intersection product X · V in A m (XV), where m = dim V — codim(X, Y). If a closed subscheme Z of XV is given, the basic problem of residual intersections is to write X · V as the sum of a class on Z and a class on a “residual set” R. There is a canonical choice for the class on Z, namely
$$\left \{ c\left ( N \right)\cap s\left ( Z,V \right) \right \}_{m}$$
where N is the restriction to Z of N X Y, and s(Z, V) is the Segre class. Our problem is therefore to compute this class on Z, and to construct and compute a residual intersection class ℝ in A m (R), for an appropriate closed set R such that ZR = XV, with
$$X\cdot V=\left \{ c\left ( N \right)\cap s\left ( Z,V \right) \right \}_{m}+\mathbb{R}$$
.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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