Residual intersection

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

Given a subset Z in the intersection of varieties, understand the complement of Z in the intersection; i.e., the residual set to Z.

The intersection determines a class , 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 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.[citation needed] 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 Kleiman (1981).

Definition

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The following definition is due to (Kleiman 1981).

Let

 

be closed embeddings, where A is an algebraic variety and Z, W are closed subschemes. Then, by definition, the residual scheme to Z is

 .

where   is the projectivization (in the classical sense) and   is the ideal sheaf defining  .

Note: if   is the blow-up of   along  , then, for  , the surjection   gives the closed embedding:

 ,

which is the isomorphism if the inclusion   is a regular embedding.

Residual intersection formula — Let  .

 

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.

If the   are scheme-theoretic connected components of  , then

 

For example, if Y is the projective space, then Bézout's theorem says the degree of   is   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   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). The intersection product of  

 

is an element of the Chow group of Y and it can be written as

 

where   are positive integers.

Given a set S, we let

 

Formulae

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Quillen's excess-intersection formula

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The formula in the topological setting is due to Quillen (1971).

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 Akd' (X″) to Akd(X″). Then

Excess intersection formula —  

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 q: P(E ⊕ 1) → X 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) = cr(ξ) is the Euler class of ξ and   is an element of Ak(P(E ⊕ 1)) that restricts to x. Since the injection q*: Akr(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. 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 e(ξ)z = 0. 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:

 

 

One special case of the formula is the self-intersection formula, which says: given a regular embedding i: XY 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   be the blow-up along a closed subscheme X,   the exceptional divisor and   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  , one gets:

Jouanolou's key formula —  .

Examples

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Throughout the example section, the base field is algebraically closed and has characteristic zero. All the examples below (except the first one) are from Fulton (1998).

Example: intersection of two plane curves containing the same component

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Let   and   be two plane curves in  . Set theoretically, their intersection

 

is the union of a point and an embedded  . By Bézout's theorem, it is expected this intersection should contain   points since it is the intersection of two conics, so interpreting this intersection requires a residual intersection. Then

   

Since   are both degree   hypersurfaces, their normal bundle is the pullback of  , hence the numerator of the two residual components is

 

Because   is given by the vanishing locus   its normal bundle is  , hence

 

since   is dimension  . Similarly, the numerator is also  , hence the residual intersection is of degree  , as expected since   is the complete intersection given by the vanishing locus  . Also, the normal bundle of   is   since it is given by the vanishing locus  , so

 

Inverting   gives the series

 

hence

 

giving the residual intersection of   for  . Pushing forward these two classes gives   in  , as desired.

Example: the degree of a curve in three surfaces

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Let   be three surfaces. Suppose the scheme-theoretic intersection   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

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The plane conics are parametrized by  . Given five general lines  , let   be the hypersurfaces of conics tangent to  ; it can be shown that these hypersurfaces have degree two.

The intersection   contains the Veronese surface   consisting of double lines; it is a scheme-theoretic connected component of  . Let   be the hyperplane class = the first Chern class of O(1) in the Chow ring of Z. Now,   such that   pulls-back to   and so the normal bundle to   restricted to Z is

 

So, the total Chern class of it is

 

Similarly, using that the normal bundle to a regular   is   as well as the Euler sequence, we get that the total Chern class of the normal bundle to   is

 

Thus, the Segre class of   is

 

Hence, the equivalence of Z is

 

By Bézout's theorem, the degree of   is   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   and  , it is:

 

Example: conics tangent to given five conics

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Suppose we are given five plane conics   in general positions. One can proceed exactly as in the previous example. Thus, let   be the hypersurface of conics tangent to  ; it can be shown that it has degree 6. The intersection   contains the Veronese surface Z of double lines.

Example: functoriality of construction of a refined Gysin homomorphism

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The fuctoriality is the section title refers to: given two regular embedding  ,

 

where the equality has the following sense:

Notes

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References

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  • Fulton, William (1998). "Chapter 9 as well as Section 17.6". Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 2 (2nd ed.). Berlin: Springer-Verlag. ISBN 978-3-540-62046-4. MR 1644323.
  • Kleiman, Steven L. (1981). "Multiple-point formulas I: Iteration". Acta Mathematica. 147 (1): 13–49. doi:10.1007/BF02392866. ISSN 0001-5962. OCLC 5655914077.
  • Quillen, Daniel (1971). "Elementary proofs of some results of cobordism theory using Steenrod operations". Advances in Mathematics. 7 (1): 29–56. doi:10.1016/0001-8708(71)90041-7. ISSN 0001-8708. OCLC 4922300265.
  • Ziv Ran, "Curvilinear enumerative geometry", Preprint, University of Chicago, 1983.

Further reading

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