Excess and Residual Intersections

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·Vas 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 $$ \mathbb{R} $$ in A m (R), for an appropriate closed set R such that Z∪R= X∩V, with $$ X \cdot V = {\left\{ {c\left( N \right) \cap s\left( {Z,V} \right)} \right\}_m} + \mathbb{R} $$ If m = 0, and R is a finite set, knowing X·V and $$ {\left\{ {c\left( N \right) \cap s\left( {Z,V} \right)} \right\}_0} $$ gives a formula for the weighted number of points of R. This is the basis for applications of the excess intersection formula to enumerative geometry. In case Z is a (scheme-theoretic) connected component of X∩V, R is the union of the other connected components; since $$ {A_ * }\left( {X \cap V} \right) = {A_ * }\left( Z \right) \oplus {A_ * }\left( R \right) $$ the above decomposition is part of the construction of Chap. 6. Computations, applications, and a few of the many classical examples are considered in § 9.1. The general case is considered in § 9.2. In the main theorem Z is assumed to be a Cartier divisor on V; in this case there is a natural scheme structure on the residual set, which can be used to construct $$ \mathbb{R} $$ If Z is arbitrary, one blows up V along Z to reduce to the divisor case. An important and typical application of the residual intersection theorem is to the formula for the double point cycle class of a morphism, which is given in § 9.3.