Residual intersection explained

In algebraic geometry, the problem of residual intersection asks the following:

Given a subset Z in the intersection

c ap ^{r} _{i =1} X _{i} of varieties, understand the complement of Z in the intersection; i.e., the residual set to Z.The intersection determines a class (X _{1} … X _{r)} , 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:(X _{1} … X _{r)} - (X _{1} … X ^{Z} _{r)} where - ^{Z} means the part supported on Z; classically the degree of the part supported on Z is called the equivalence of Z.

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. The modern formulation of the problems and the solutions is due to Fulton and MacPherson. To be precise, they develop the intersection theory by a way of solving the problems of residual intersections (namely, by the use of the Segre class of a normal cone to an intersection.) A generalization to a situation where the assumption on regular embedding is weakened is due to .

Definition

The following definition is due to .

Let

Z \subset W \subset A be closed embeddings, where A is an algebraic variety and Z, W are closed subschemes. Then, by definition, the residual scheme to Z isR(Z, W) = P (I(Z, W) ^{*)} .where P is the projectivization (in the classical sense) and l{I}(Z, W) \subset l{O} _{W} is the ideal sheaf defining Z \hookrightarrow W .

Note: if

B(Z, W) is the blow-up of W along Z , then, for l{I} = l{I}(Z, W) , the surjection $\backslash operatorname(\backslash mathcal)\; \backslash to\; \backslash bigoplus\_^\; \backslash mathcal^n$ gives the closed embedding:B(Z, W) \hookrightarrow R(Z, W) ,which is the isomorphism if the inclusion Z \hookrightarrow W is a regular embedding.

If the

Z _{i} are scheme-theoretic connected components of c ap _{i} X _{i} , then(X _{1} … X _{r)} = \sum _{i} (X _{1} … X ^{Z _{i}} _{r)}

For example, if Y is the projective space, then Bézout's theorem says the degree of

c ap _{i} X _{i} is \prod _{i} \deg(X _{i)} and so the above is a different way to count the contributions to the degree of the intersection. In fact, in applications, one combines Bézout's theorem.

Let

X _{i} \hookrightarrow Y be regular embeddings of schemes, separated and of finite type over the base field; for example, this is the case if X_i are effective Cartier divisors (e.g., hypersurfaces). The intersection product of X _{i} (X _{1} … X _{r)} is an element of the Chow group of Y and it can be written as(X _{1} … X _{r)} = \sum _{i} m _{i} \alpha _{i} where m _{i} are positive integers.

Given a set S, we let

(X _{1} … X ^{S} _{r)} = \sum _{\operatorname{Supp} (\alpha _{i)} \subset S} m _{i} \alpha _{i.}

Formulae

Quillen's excess-intersection formula

The formula in the topological setting is due to .

Now, suppose we are given Y → Y and suppose i: X = X ×_Y Y → Y is regular of codimension d so that one can define i^! as before. Let F be the excess bundle of i and i^'; that is, it is the pullback to X of the quotient of N by the normal bundle of i. Let e(F) be the Euler class (top Chern class) of F, which we view as a homomorphism from A_k−d (X) to A_k−d(X). Then

where i^! is determined by the morphism Y → Y → Y.

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. 17 of loc. cit.

Proof: One can deduce the intersection formula from the rather explicit form of a Gysin homomorphism. Let E be a vector bundle on X of rank r and the projective bundle (here 1 means the trivial line bundle). As usual, we identity P(E ⊕ 1) as a disjoint union of P(E) and E. Then there is the tautological exact sequence

0 \to l{O}(-1) \to q ^{*E} ⊕ 1 \to \xi \to 0 on P(E ⊕ 1). We claim the Gysin homomorphism is given asA _{k(E)} \to A _{k-r} (X), x \mapsto q _{*(e(\xi)} \overline{x}) where e(ξ) = c_r(ξ) is the Euler class of ξ and \overline{x} is an element of that restricts to x. Since the injection splits, we can write\overline{x} = q ^{*} y + z 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 soe(\xi) = \sum ^{r} _{0} c ^{i} _{1 ( l{O}(1))} c _{r-i} (q ^{*} E). Then we get:q _{*(e(\xi)} q ^{*} y) = \sum ^{r} _{i =0} s _{i-r} (E ⊕ 1 ) c _{r-i} (E) y where s_I(E ⊕ 1) is the i-th Segre class. Since the zeroth term of a Segre class is the identity and its negative terms are zero, the above expression equals y. Next, since the restriction of ξ to P(E) has a nowhere-vanishing section and z is a class of a cycle supported on P(E), it follows that