Intersection theory
Template:Short description Template:Distinguish Template: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.Template: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.Template:Sfn
Topological intersection form
Template:See also For a connected oriented manifold Template:Mvar of dimension Template:Math the intersection form is defined on the Template:Mvar-th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class Template:Math in Template:Math. Stated precisely, there is a bilinear form
given by
with
This is a symmetric form for Template:Mvar even (so Template:Math doubly even), in which case the signature of Template:Mvar is defined to be the signature of the form, and an alternating form for Template:Mvar odd (so Template:Math is singly even). These can be referred to uniformly as ε-symmetric forms, where Template:Math respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an [[ε-quadratic form|Template: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 Template: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 Template:Mvar-dimensional submanifolds Template:Mvar, Template:Mvar for the Poincaré duals of Template:Mvar and Template:Mvar. Then Template:Math is the oriented intersection number of Template:Mvar and Template:Mvar, which is well-defined because since dimensions of Template:Mvar and Template:Mvar sum to the total dimension of Template: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 Template:Mvar and Template:Mvar are subvarieties of a non-singular variety Template:Mvar, the intersection product Template:Math should be an equivalence class of algebraic cycles closely related to the geometry of how Template:Math, Template:Mvar and Template:Mvar are situated in Template:Mvar. Two extreme cases have been most familiar. If the intersection is proper, i.e. Template:Math, then Template:Math is a linear combination of the irreducible components of Template:Math, with coefficients the intersection multiplicities. At the other extreme, if Template:Math is a non-singular subvariety, the self-intersection formula says that Template:Math is represented by the top Chern class of the normal bundle of Template:Mvar in Template: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 Template:Mvar and Template:Mvar requires more than taking just the set-theoretic intersection Template:Math of the cycles in question. If the two cycles are in "good position" then the intersection product, denoted Template: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 Template:Mvar and Template:Mvar is called proper if the codimension of the (set-theoretic) intersection Template:Math is the sum of the codimensions of Template:Mvar and Template: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 Template:Mvar and Template:Mvar, there are equivalent cycles Template:Math and Template:Math such that the intersection Template:Math is proper. Of course, on the other hand, for a second equivalent Template:Math and Template:Math, Template:Math needs to be equivalent to Template:Math.
For the purposes of intersection theory, rational equivalence is the most important one. Briefly, two Template:Mvar-dimensional cycles on a variety Template:Mvar are rationally equivalent if there is a rational function Template:Math on a Template:Math-dimensional subvariety Template:Mvar, i.e. an element of the function field Template:Math or equivalently a function Template:Math, such that Template:Math, where Template: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 Template:Math and an axis Template:Math should be Template:Math, because if one of the cycles moves (yet in an undefined sense), there are precisely two intersection points which both converge to Template:Math when the cycles approach the depicted position. (The picture is misleading insofar as the apparently empty intersection of the parabola and the line Template: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 Template:Mvar be smooth (or all local rings regular). Further let Template:Mvar and Template: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 Template:Mvar and Template:Mvar in the coordinate ring of Template:Mvar. Let Template:Mvar be an irreducible component of the set-theoretic intersection Template:Math and Template:Mvar its generic point. The multiplicity of Template:Mvar in the intersection product Template:Math is defined by
the alternating sum over the length over the local ring of Template:Mvar in Template: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
- 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 Template:Mvar and Template: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 Template:Math.
The Chow ring
Template:Main The Chow ring is the group of algebraic cycles modulo rational equivalence together with the following commutative intersection product:
whenever V and W meet properly, where is the decomposition of the set-theoretic intersection into irreducible components.
Self-intersection
Given two subvarieties Template:Mvar and Template:Mvar, one can take their intersection Template:Math, but it is also possible, though more subtle, to define the self-intersection of a single subvariety.
Given, for instance, a curve Template:Mvar on a surface Template:Mvar, its intersection with itself (as sets) is just itself: Template: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: Template:Math, while self-intersection is like squaring a single number: Template: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 Template:Mvar not with itself, but with a slightly pushed off version of itself. In the plane, this just means translating the curve Template:Mvar in some direction, but in general one talks about taking a curve Template:Math that is linearly equivalent to Template:Mvar, and counting the intersection Template:Math, thus obtaining an intersection number, denoted Template:Math. Note that unlike for distinct curves Template:Mvar and Template:Mvar, the actual points of intersection are not defined, because they depend on a choice of Template:Math, but the “self intersection points of Template:Math can be interpreted as Template:Mvar generic points on Template:Mvar, where Template:Math. More properly, the self-intersection point of Template:Mvar is the generic point of Template:Mvar, taken with multiplicity Template:Math.
Alternatively, one can “solve” (or motivate) this problem algebraically by dualizing, and looking at the class of Template: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 Template:Mvar in the projective plane Template:Math: it has self-intersection number 1 since all other lines cross it once: one can push Template:Mvar off to Template:Math, and Template:Math (for any choice) of Template:Math, hence Template:Math. In terms of intersection forms, we say the plane has one of type Template: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 Template: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 Template:Math (which can also be interpreted as the non-singular quadric Template:Mvar in Template:Math) has self-intersection Template:Math, since a line can be moved off itself. (It is a ruled surface.) In terms of intersection forms, we say Template:Math has one of type Template:Mvar – there are two basic classes of lines, which intersect each other in one point (Template:Mvar), but have zero self-intersection (no Template:Math or Template: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 Template:Mvar, blowing up at a point creates a curve Template:Mvar. This curve Template:Mvar is recognisable by its genus, which is Template:Math, and its self-intersection number, which is Template:Math. (This is not obvious.) Note that as a corollary, Template:Math and Template: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 Template:Math-curve is the exceptional curve of some blow-up (it can be “blown down”).