, the intersection product, in the Chow group of an ambient space and, in this situation, the problem is to understand the class, the residual class to Z: where
The two principal applications are the solutions to problems in enumerative geometry (e.g., Steiner's conic problem) and the derivation of the multiple-point formula, the formula allowing one to count or enumerate the points in a fiber even when they are infinitesimally close.
The problem of residual intersection goes back to the 19th century.
[citation needed] The modern formulation of the problems and the solutions is due to Fulton and MacPherson.
A generalization to a situation where the assumption on regular embedding is weakened is due to Kleiman (1981).
gives the closed embedding: which is the isomorphism if the inclusion
where s(CZ X) denotes the Segre class of the normal cone to Z in X and the subscript Z signifies the part supported on Z.
, then For example, if Y is the projective space, then Bézout's theorem says the degree of
In fact, in applications, one combines Bézout's theorem.
be regular embeddings of schemes, separated and of finite type over the base field; for example, this is the case if Xi are effective Cartier divisors (e.g., hypersurfaces).
Finally, it is possible to generalize the above construction and formula to complete intersection morphisms; this extension is discussed in § 6.6. as well as Ch.
Proof: One can deduce the intersection formula from the rather explicit form of a Gysin homomorphism.
As usual, we identity P(E ⊕ 1) as a disjoint union of P(E) and E. Then there is the tautological exact sequence on P(E ⊕ 1).
Since the injection q*: Ak−r(X) → Ak(P(E ⊕ 1)) splits, we can write where z is a class of a cycle supported on P(E).
By the Whitney sum formula, we have: c(q*E) = (1 − c1(O(1)))c(ξ) and so Then we get: where sI(E ⊕ 1) is the i-th Segre class.
Hence, writing π for the projection map of E and j for the inclusion E to P(E⊕1), we get: where the second-to-last equality is because of the support reason as before.
This completes the proof of the explicit form of the Gysin homomorphism.
the restriction of f. Assume f can be written as a closed immersion followed by a smooth morphism (for example, Y is quasi-projective).
Throughout the example section, the base field is algebraically closed and has characteristic zero.
Pushing forward these two classes gives
is the disjoint union of a smooth curve C and a zero-dimensional schem S. One can ask: what is the degree of S?
consisting of double lines; it is a scheme-theoretic connected component of
restricted to Z is So, the total Chern class of it is Similarly, using that the normal bundle to a regular
as well as the Euler sequence, we get that the total Chern class of the normal bundle to
is Hence, the equivalence of Z is By Bézout's theorem, the degree of
and hence the residual set consists of a single point corresponding to a unique conic tangent to the given all five lines.
Alternatively, the equivalence of Z can be computed by #formula?
One can proceed exactly as in the previous example.
contains the Veronese surface Z of double lines.
The fuctoriality is the section title refers to: given two regular embedding