Английская Википедия:Intersection theory
Шаблон:Short description Шаблон:Distinguish Шаблон:No footnotes In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety.Шаблон:Sfn The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form.
There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov–Witten theory and the extension of intersection theory from schemes to stacks.Шаблон:Sfn
Topological intersection form
Шаблон:See also For a connected oriented manifold Шаблон:Mvar of dimension Шаблон:Math the intersection form is defined on the Шаблон:Mvar-th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class Шаблон:Math in Шаблон:Math. Stated precisely, there is a bilinear form
- <math>\lambda_M \colon H^n(M,\partial M) \times H^n(M,\partial M)\to \mathbf{Z}</math>
given by
- <math>\lambda_M(a,b)=\langle a\smile b,[M]\rangle \in \mathbf{Z}</math>
with
- <math>\lambda_M(a,b)=(-1)^n\lambda_M(b,a) \in \mathbf{Z}.</math>
This is a symmetric form for Шаблон:Mvar even (so Шаблон:Math doubly even), in which case the signature of Шаблон:Mvar is defined to be the signature of the form, and an alternating form for Шаблон:Mvar odd (so Шаблон:Math is singly even). These can be referred to uniformly as ε-symmetric forms, where Шаблон:Math respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an [[ε-quadratic form|Шаблон:Mvar-quadratic form]], though this requires additional data such as a framing of the tangent bundle. It is possible to drop the orientability condition and work with Шаблон:Math coefficients instead.
These forms are important topological invariants. For example, a theorem of Michael Freedman states that simply connected compact 4-manifolds are (almost) determined by their intersection forms up to homeomorphism.
By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative Шаблон:Mvar-dimensional submanifolds Шаблон:Mvar, Шаблон:Mvar for the Poincaré duals of Шаблон:Mvar and Шаблон:Mvar. Then Шаблон:Math is the oriented intersection number of Шаблон:Mvar and Шаблон:Mvar, which is well-defined because since dimensions of Шаблон:Mvar and Шаблон:Mvar sum to the total dimension of Шаблон:Mvar they generically intersect at isolated points. This explains the terminology intersection form.
Intersection theory in algebraic geometry
William Fulton in Intersection Theory (1984) writes
... if Шаблон:Mvar and Шаблон:Mvar are subvarieties of a non-singular variety Шаблон:Mvar, the intersection product Шаблон:Math should be an equivalence class of algebraic cycles closely related to the geometry of how Шаблон:Math, Шаблон:Mvar and Шаблон:Mvar are situated in Шаблон:Mvar. Two extreme cases have been most familiar. If the intersection is proper, i.e. Шаблон:Math, then Шаблон:Math is a linear combination of the irreducible components of Шаблон:Math, with coefficients the intersection multiplicities. At the other extreme, if Шаблон:Math is a non-singular subvariety, the self-intersection formula says that Шаблон:Math is represented by the top Chern class of the normal bundle of Шаблон:Mvar in Шаблон:Mvar.
To give a definition, in the general case, of the intersection multiplicity was the major concern of André Weil's 1946 book Foundations of Algebraic Geometry. Work in the 1920s of B. L. van der Waerden had already addressed the question; in the Italian school of algebraic geometry the ideas were well known, but foundational questions were not addressed in the same spirit.
Moving cycles
A well-working machinery of intersecting algebraic cycles Шаблон:Mvar and Шаблон:Mvar requires more than taking just the set-theoretic intersection Шаблон:Math of the cycles in question. If the two cycles are in "good position" then the intersection product, denoted Шаблон:Math, should consist of the set-theoretic intersection of the two subvarieties. However cycles may be in bad position, e.g. two parallel lines in the plane, or a plane containing a line (intersecting in 3-space). In both cases the intersection should be a point, because, again, if one cycle is moved, this would be the intersection. The intersection of two cycles Шаблон:Mvar and Шаблон:Mvar is called proper if the codimension of the (set-theoretic) intersection Шаблон:Math is the sum of the codimensions of Шаблон:Mvar and Шаблон:Mvar, respectively, i.e. the "expected" value.
Therefore, the concept of moving cycles using appropriate equivalence relations on algebraic cycles is used. The equivalence must be broad enough that given any two cycles Шаблон:Mvar and Шаблон:Mvar, there are equivalent cycles Шаблон:Math and Шаблон:Math such that the intersection Шаблон:Math is proper. Of course, on the other hand, for a second equivalent Шаблон:Math and Шаблон:Math, Шаблон:Math needs to be equivalent to Шаблон:Math.
For the purposes of intersection theory, rational equivalence is the most important one. Briefly, two Шаблон:Mvar-dimensional cycles on a variety Шаблон:Mvar are rationally equivalent if there is a rational function Шаблон:Math on a Шаблон:Math-dimensional subvariety Шаблон:Mvar, i.e. an element of the function field Шаблон:Math or equivalently a function Шаблон:Math, such that Шаблон:Math, where Шаблон:Math is counted with multiplicities. Rational equivalence accomplishes the needs sketched above.
Intersection multiplicities
The guiding principle in the definition of intersection multiplicities of cycles is continuity in a certain sense. Consider the following elementary example: the intersection of a parabola Шаблон:Math and an axis Шаблон:Math should be Шаблон:Math, because if one of the cycles moves (yet in an undefined sense), there are precisely two intersection points which both converge to Шаблон:Math when the cycles approach the depicted position. (The picture is misleading insofar as the apparently empty intersection of the parabola and the line Шаблон:Math is empty, because only the real solutions of the equations are depicted).
The first fully satisfactory definition of intersection multiplicities was given by Serre: Let the ambient variety Шаблон:Mvar be smooth (or all local rings regular). Further let Шаблон:Mvar and Шаблон:Mvar be two (irreducible reduced closed) subvarieties, such that their intersection is proper. The construction is local, therefore the varieties may be represented by two ideals Шаблон:Mvar and Шаблон:Mvar in the coordinate ring of Шаблон:Mvar. Let Шаблон:Mvar be an irreducible component of the set-theoretic intersection Шаблон:Math and Шаблон:Mvar its generic point. The multiplicity of Шаблон:Mvar in the intersection product Шаблон:Math is defined by
- <math>\mu(Z; V, W) := \sum^\infty_{i=0} (-1)^i \text{length}_{\mathcal O_{X, z}} \text{Tor}_i^{\mathcal O_{X, z}} (\mathcal O_{X, z}/I, \mathcal O_{X, z}/J),</math>
the alternating sum over the length over the local ring of Шаблон:Mvar in Шаблон:Mvar of torsion groups of the factor rings corresponding to the subvarieties. This expression is sometimes referred to as Serre's Tor-formula.
Remarks:
- The first summand, the length of
- <math> \left ( \mathcal O_{X, z}/I \right ) \otimes_{\mathcal O_{X, z}} \left (\mathcal O_{X, z}/J \right ) = \mathcal O_{Z, z}</math>
- is the "naive" guess of the multiplicity; however, as Serre shows, it is not sufficient.
- The sum is finite, because the regular local ring <math>\mathcal O_{X, z}</math> has finite Tor-dimension.
- If the intersection of Шаблон:Mvar and Шаблон:Mvar is not proper, the above multiplicity will be zero. If it is proper, it is strictly positive. (Both statements are not obvious from the definition).
- Using a spectral sequence argument, it can be shown that Шаблон:Math.
The Chow ring
Шаблон:Main The Chow ring is the group of algebraic cycles modulo rational equivalence together with the following commutative intersection product:
- <math>V \cdot W := \sum_{i} \mu(Z_i; V, W)Z_i</math>
whenever V and W meet properly, where <math>V \cap W = \cup_i Z_i</math> is the decomposition of the set-theoretic intersection into irreducible components.
Self-intersection
Given two subvarieties Шаблон:Mvar and Шаблон:Mvar, one can take their intersection Шаблон:Math, but it is also possible, though more subtle, to define the self-intersection of a single subvariety.
Given, for instance, a curve Шаблон:Mvar on a surface Шаблон:Mvar, its intersection with itself (as sets) is just itself: Шаблон:Math. This is clearly correct, but on the other hand unsatisfactory: given any two distinct curves on a surface (with no component in common), they intersect in some set of points, which for instance one can count, obtaining an intersection number, and we may wish to do the same for a given curve: the analogy is that intersecting distinct curves is like multiplying two numbers: Шаблон:Math, while self-intersection is like squaring a single number: Шаблон:Math. Formally, the analogy is stated as a symmetric bilinear form (multiplication) and a quadratic form (squaring).
A geometric solution to this is to intersect the curve Шаблон:Mvar not with itself, but with a slightly pushed off version of itself. In the plane, this just means translating the curve Шаблон:Mvar in some direction, but in general one talks about taking a curve Шаблон:Math that is linearly equivalent to Шаблон:Mvar, and counting the intersection Шаблон:Math, thus obtaining an intersection number, denoted Шаблон:Math. Note that unlike for distinct curves Шаблон:Mvar and Шаблон:Mvar, the actual points of intersection are not defined, because they depend on a choice of Шаблон:Math, but the “self intersection points of Шаблон:Math can be interpreted as Шаблон:Mvar generic points on Шаблон:Mvar, where Шаблон:Math. More properly, the self-intersection point of Шаблон:Mvar is the generic point of Шаблон:Mvar, taken with multiplicity Шаблон:Math.
Alternatively, one can “solve” (or motivate) this problem algebraically by dualizing, and looking at the class of Шаблон:Math – this both gives a number, and raises the question of a geometric interpretation. Note that passing to cohomology classes is analogous to replacing a curve by a linear system.
Note that the self-intersection number can be negative, as the example below illustrates.
Examples
Consider a line Шаблон:Mvar in the projective plane Шаблон:Math: it has self-intersection number 1 since all other lines cross it once: one can push Шаблон:Mvar off to Шаблон:Math, and Шаблон:Math (for any choice) of Шаблон:Math, hence Шаблон:Math. In terms of intersection forms, we say the plane has one of type Шаблон:Math (there is only one class of lines, and they all intersect with each other).
Note that on the affine plane, one might push off Шаблон:Mvar to a parallel line, so (thinking geometrically) the number of intersection points depends on the choice of push-off. One says that “the affine plane does not have a good intersection theory”, and intersection theory on non-projective varieties is much more difficult.
A line on a Шаблон:Math (which can also be interpreted as the non-singular quadric Шаблон:Mvar in Шаблон:Math) has self-intersection Шаблон:Math, since a line can be moved off itself. (It is a ruled surface.) In terms of intersection forms, we say Шаблон:Math has one of type Шаблон:Mvar – there are two basic classes of lines, which intersect each other in one point (Шаблон:Mvar), but have zero self-intersection (no Шаблон:Math or Шаблон:Math terms).
Blow-ups
A key example of self-intersection numbers is the exceptional curve of a blow-up, which is a central operation in birational geometry. Given an algebraic surface Шаблон:Mvar, blowing up at a point creates a curve Шаблон:Mvar. This curve Шаблон:Mvar is recognisable by its genus, which is Шаблон:Math, and its self-intersection number, which is Шаблон:Math. (This is not obvious.) Note that as a corollary, Шаблон:Math and Шаблон:Math are minimal surfaces (they are not blow-ups), since they do not have any curves with negative self-intersection. In fact, Castelnuovo’s contraction theorem states the converse: every Шаблон:Math-curve is the exceptional curve of some blow-up (it can be “blown down”).
See also
Citations
References
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