Excess and Residual Intersections

Summary

If X ↪ Y is a regular imbedding, V ⊂ Y a subvariety, we have constructed (§ 6.1) an intersection product X · V in A _m (X ∩ V), where m = dim V — codim(X, Y). If a closed subscheme Z of X ∩ V 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 Z ∪ R = X ∩ V, with

$$X\cdot V=\left \{ c\left ( N \right)\cap s\left ( Z,V \right) \right \}_{m}+\mathbb{R}$$

.