Residual intersection

In algebraic geometry, the problem of residual intersection asks the following:
The intersection determines a class formula_2, 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 formula_4 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 .
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
on P("E" ⊕ 1). We claim the Gysin homomorphism is given as
where "e"("ξ") = "c""r"("ξ") is the Euler class of ξ and formula_7 is an element of that restricts to "x". Since the injection 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 − "c"1("O"(1)))"c"("ξ") and so
Then we get:
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 . 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 rest is formal and straightforward. We use the exact sequence
where "r" is the projection map for . Writing "P" for the closure of the specialization of "V", by the Whitney sum formula and the projection formula, we have:
formula_14
One special case of the formula is the self-intersection formula, which says: given a regular embedding "i": "X" → "Y" with normal bundle "N",
(To get this, take "Y" = "Y" = "X".) For example, from this and the projection formula, when "X", "Y" are smooth, one can deduce the formula:
in the Chow ring of "Y".
Let formula_17 be the blow-up along a closed subscheme "X", formula_18 the exceptional divisor and formula_19 the restriction of "f". Assume "f" can be written as a closed immersion followed by a smooth morphism (for example, "Y" is quasi-projective). Then, from formula_20, one gets:
Examples.
Throughout the example section, the base field is algebraically closed and has characteristic zero. All the examples below (except the first one) are from .
Example: intersection of two plane curves containing the same component.
Let formula_21 and formula_22 be two plane curves in formula_23. Set theoretically, their intersectionformula_24is the union of a point and an embedded formula_25. By Bézout's theorem, it is expected this intersection should contain formula_26 points since it is the intersection of two conics, so interpreting this intersection requires a residual intersection. Thenformula_27
formula_28Since formula_29 are both degree formula_30 hypersurfaces, their normal bundle is the pullback of formula_31, hence the numerator of the two residual components isformula_32Because formula_33 is given by the vanishing locus formula_34 its normal bundle is formula_35, henceformula_36since formula_33 is dimension formula_38. Similarly, the numerator is also formula_39, hence the residual intersection is of degree formula_39, as expected since formula_33 is the complete intersection given by the vanishing locus formula_34. Also, the normal bundle of formula_43 is formula_44 since it's given by the vanishing locus formula_45, soformula_46Inverting formula_47 gives the seriesformula_48hence formula_49giving the residual intersection of formula_50 for formula_43. Pushing forward these two classes gives formula_52 in formula_53, as desired.
Example: the degree of a curve in three surfaces.
Let formula_54 be three surfaces. Suppose the scheme-theoretic intersection formula_55 is the disjoint union of a smooth curve "C" and a zero-dimensional schem "S". One can ask: what is the degree of "S"? This can be answered by #formula.
Example: conics tangent to given five lines.
The plane conics are parametrized by formula_56. Given five general lines formula_57, let formula_58 be the hypersurfaces of conics tangent to formula_59; it can be shown that these hypersurfaces have degree two.
The intersection formula_60 contains the Veronese surface formula_61 consisting of double lines; it is a scheme-theoretic connected component of formula_60. Let formula_63
be the hyperplane class = the first Chern class of "O"(1) in the Chow ring of "Z". Now, formula_64 such that formula_65 pulls-back to formula_66 and so the normal bundle to formula_67 restricted to "Z" is
So, the total Chern class of it is
Similarly, using that the normal bundle to a regular formula_70 is formula_71 as well as the Euler sequence, we get that the total Chern class of the normal bundle to formula_72 is
Thus, the Segre class of formula_72 is
Hence, the equivalence of "Z" is
By Bézout's theorem, the degree of formula_60 is formula_78 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?; since formula_79 and formula_80, it is:
Example: conics tangent to given five conics.
Suppose we are given five plane conics formula_82 in general positions. One can proceed exactly as in the previous example. Thus, let formula_83 be the hypersurface of conics tangent to formula_84; it can be shown that it has degree 6. The intersection formula_85 contains the Veronese surface "Z" of double lines.
Example: functoriality of construction of a refined Gysin homomorphism.
The fuctoriality is the section title refers to: given two regular embedding formula_86,
where the equality has the following sense: