In this post, we will take on intersection theory – which is pretty much just what it sounds like, except for a few modifications which we will later discuss. For example, we may ask, where do the curves (a parabola) and the line (a horizontal line) intersect? It only takes a little high school algebra and analytic geometry (which is really a more elementary form of what we now more properly call algebraic geometry) to find that they intersect at the two points and .
Suppose, instead, that we want to take the intersection of the parabola and the horizontal line . If the coordinates and are real numbers, we would have no intersection. But if they are complex numbers, then we will find that they do intersect, once again at two points, namely and . When complex numbers are involved, it may be difficult to visualize things – for example, the complex numbers are often visualized as a plane, not as a line – but we will continue to refer, say, to as a “line”. This is rather common practice in algebraic geometry – recall that we have also been referring to a torus as an elliptic curve (see The Moduli Space of Elliptic Curves)!
Consider now the intersection of the parabola and the horizontal line . This time, contrary to the earlier two cases, the curves intersect only at one point, namely at . But we would like to think of them as intersecting “twice”, even though the intersection occurs only at a single point. Hence, we say that the point has intersection multiplicity equal to .
The notion of intersection multiplicities make sense – generally speaking, for instance, imagining a random parabola and a random line in the – plane, they will generally intersect at two points, except in certain instances, such as when the line is tangent to the parabola – this is of course the special, less general, case. In order to make our counting of intersections consistent, even with these special cases, we need this idea of “intersection multiplicities.”
Another way to think of the previous example is that the parabola and the line having only one intersection is such a special case that simply “displacing” or “moving” either curve by a little bit results in them having two intersections. Consider, for example, the following diagram courtesy of user Jakob.scholbach of Wikipedia:
Equipped with the idea of intersection multiplicities (which we will explicitly give the formula for later), we have Bezout’s theorem, which states that the number of intersections of two curves, counted with their intersection multiplicities, is equal to the product of the degrees of the polynomials that define them. For example, two parabolas will generally intersect at four points, except in special cases where their intersections have multiplicities greater than .
For higher-dimensional varieties, the intersections need not be points, but other kinds of varieties. For an -dimensional variety embedded in a larger -dimensional variety (which we may think of as the space the -dimensional variety is living in), the codimension of in is given by . If the codimension of the intersection of two varieties is equal to the sum of the codimensions of the intersecting varieties, then we say that the intersection is proper.
Proper intersections correspond to our intuition. For example, consider again curves such as parabolas and lines in the plane. The plane is -dimensional, while curves are dimensional. Therefore their codimension in the plane is equal to . Proper intersections will then be points, which have dimension equal to and therefore have codimension in the plane equal to . Similarly, the proper intersection of two surfaces, for example two planes, in some -dimensional space, is a curve (a line in the case of two planes), since surfaces have codimension equal to inside the -dimensional space, while curves have codimension equal to .
We can now give the definition of the intersection multiplicity. It is quite technical, involving the Tor functor (see The Hom and Tensor Functors), but we will also give the special case for curves, which is a little less technical compared to the general case. Let and be two subvarieties of some smooth variety (for a discussion of smoothness and singularities see Reduction of Elliptic Curves Modulo Primes) which intersect properly and let be their set-theoretic intersection. Then the intersection multiplicity is given by
where and are the ideals corresponding to the varieties and respectively, and is the generic point of the variety (we are using here the definition of a variety as a scheme satisfying certain conditions, which we have not actually discussed in this blog yet – we will leave this to the references for now). The concept of length is a generalization of the concept of dimension in algebraic geometry, and refers to the length (the ordinary use of the term) of the longest chain of modules that contain one another (while dimension refers to the length of the longest chain of rings that contain one another).
If and are curves on a surface then the above formula reduces to
Another concept in algebraic geometry closely related to intersection theory is that of an algebraic cycle. Algebraic cycles generalize the idea of divisors (see Divisors and the Picard Group). Algebraic cycles on a variety can be thought of as “linear combinations” of the subvarieties (satisfying certain conditions, such as being closed, reduced, and irreducible, so that they are not unions of other subvarieties) on . Divisors themselves are just algebraic cycles of codimension ; in other words, they are algebraic cycles whose dimension is less than the variety in which they are embedded. In Divisors and the Picard Group, we considered curves, which are varieties of dimension , hence the divisors on the curves were linear combinations (with integer coefficients) of points, i.e. subvarieties of dimension equal to .
Analogous to the Picard group for divisors we have the Chow group of algebraic cycles modulo rational equivalence. Two algebraic cycles and on a variety are said to be rationally equivalent if there is a rational function such that , counting multiplicities. Chow’s moving lemma states that for any two algebraic cycles and on a smooth, quasi-projective (quasi-projective means it is the intersection of a Zariski-open and Zariski-closed subset in some projective space) variety , there exists another algebraic cycle rationally equivalent to such that and intersect properly. Besides rational equivalence, there are also other notions of equivalence for algebraic cycles, such as algebraic, homological, and numerical equivalence; all of these are important in the study of algebraic cycles, but we will leave them to the references for now.
Taking intersections of subvarieties gives the Chow group a ring structure (we therefore have the concept of an intersection product). In this context we may also refer to the Chow group as the Chow ring. The Chow ring is also an example of a graded ring, which means that the intersection product is a mapping that sends a pair of equivalence classes of algebraic cycles, one with codimension and another with codimension , to an equivalence class of algebraic cycles with codimension :
The intersection product carries over into cohomology, corresponding to the so-called cup product of cohomology classes. Actually, there are many cohomology theories, but the ones considered to be “good” cohomology theories (more technically, they are the ones referred to as the Weil cohomology theories) are required to have a cycle map. Related to the notion of the cycle map is the famous Hodge conjecture in complex algebraic geometry, which states that under a certain well-known decomposition of the cohomology groups , all cohomology classes of a certain kind (the so-called Hodge classes) come from algebraic cycles. Another similar conjecture is the Tate conjecture, which relates the cohomology classes coming from algebraic cycles to the elements that are fixed by the action of the Galois group (see Galois Groups). Other important conjectures in the study of algebraic cycles are the so-called standard conjectures formulated by Alexander Grothendieck as part of his strategy to prove the Weil conjectures (see The Riemann Hypothesis for Curves over Finite Fields). The Weil conjectures were proved without the need to prove the standard conjectures, but the standard conjectures themselves continue to be the object of modern mathematical research.
Algebraic Geometry by Robin Hartshorne