Intersection theory
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In
There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings,
Topological intersection form
For a
given by
with
This is a
These forms are important
By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative n-dimensional submanifolds A, B for the Poincaré duals of a and b. Then λM (a, b) is the oriented intersection number of A and B, which is well-defined because since dimensions of A and B sum to the total dimension of M 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 A and B are subvarieties of a non-singular variety X, the intersection product A · B should be an equivalence class of algebraic cycles closely related to the geometry of how A ∩ B, A and B are situated in X. Two extreme cases have been most familiar. If the intersection is proper, i.e. dim(A ∩ B) = dim A + dim B − dim X, then A · B is a linear combination of the irreducible components of A ∩ B, with coefficients the intersection multiplicities. At the other extreme, if A = B is a non-singular subvariety, the self-intersection formula says that A · B is represented by the top Chern class of the normal bundle of A in X.
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 V and W requires more than taking just the set-theoretic intersection V ∩ W of the cycles in question. If the two cycles are in "good position" then the intersection product, denoted V · W, 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 V and W is called proper if the codimension of the (set-theoretic) intersection V ∩ W is the sum of the codimensions of V and W, respectively, i.e. the "expected" value.
Therefore, the concept of moving cycles using appropriate
For the purposes of intersection theory, rational equivalence is the most important one. Briefly, two r-dimensional cycles on a variety X are rationally equivalent if there is a rational function f on a (r + 1)-dimensional subvariety Y, i.e. an element of the function field k(Y) or equivalently a function f : Y → P1, such that V − W = f −1(0) − f −1(∞), where f −1(⋅) is counted with multiplicities. Rational equivalence accomplishes the needs sketched above.
Intersection multiplicities
The guiding principle in the definition of
The first fully satisfactory definition of intersection multiplicities was given by Serre: Let the ambient variety X be smooth (or all local rings regular). Further let V and W 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 I and J in the coordinate ring of X. Let Z be an irreducible component of the set-theoretic intersection V ∩ W and z its generic point. The multiplicity of Z in the intersection product V · W is defined by
the alternating sum over the length over the local ring of X in z 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
- is the "naive" guess of the multiplicity; however, as Serre shows, it is not sufficient.
- The sum is finite, because the regular local ring has finite Tor-dimension.
- If the intersection of V and W 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 μ(Z; V, W) = μ(Z; W, V).
The Chow ring
The
whenever V and W meet properly, where is the decomposition of the set-theoretic intersection into irreducible components.
Self-intersection
Given two subvarieties V and W, one can take their intersection V ∩ W, but it is also possible, though more subtle, to define the self-intersection of a single subvariety.
Given, for instance, a curve C on a surface S, its intersection with itself (as sets) is just itself: C ∩ C = C. 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: xy, while self-intersection is like squaring a single number: x2. 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 C not with itself, but with a slightly pushed off version of itself. In the plane, this just means translating the curve C in some direction, but in general one talks about taking a curve C′ that is linearly equivalent to C, and counting the intersection C · C′, thus obtaining an intersection number, denoted C · C. Note that unlike for distinct curves C and D, the actual points of intersection are not defined, because they depend on a choice of C′, but the “self intersection points of C′′ can be interpreted as k generic points on C, where k = C · C. More properly, the self-intersection point of C is the generic point of C, taken with multiplicity C · C.
Alternatively, one can “solve” (or motivate) this problem algebraically by dualizing, and looking at the class of [C] ∪ [C] – 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 L in the projective plane P2: it has self-intersection number 1 since all other lines cross it once: one can push L off to L′, and L · L′ = 1 (for any choice) of L′, hence L · L = 1. In terms of intersection forms, we say the plane has one of type x2 (there is only one class of lines, and they all intersect with each other).
Note that on the affine plane, one might push off L 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 P1 × P1 (which can also be interpreted as the non-singular quadric Q in P3) has self-intersection 0, since a line can be moved off itself. (It is a ruled surface.) In terms of intersection forms, we say P1 × P1 has one of type xy – there are two basic classes of lines, which intersect each other in one point (xy), but have zero self-intersection (no x2 or y2 terms).
Blow-ups
A key example of self-intersection numbers is the exceptional curve of a blow-up, which is a central operation in
See also
Citations
- ^ Eisenbud & Harris 2016, p. 14.
- ^ Eisenbud & Harris 2016, p. 2.
References
- Gathman, Andreas, Algebraic Geometry, archived from the original on 2016-05-21, retrieved 2018-05-11
- Tian, Yichao, Course Notes in Intersection Theory (PDF)[dead link]
Bibliography
- ISBN 978-1-107-01708-5.
- MR1644323
- Fulton, William; ISBN 978-1-4419-3073-6
- MR 0201468