Intersection theory

on an algebraic variety

The theory of intersections of algebraic subvarieties and cycles. Let be a smooth algebraic variety of dimension over a field , while and are subvarieties of of codimension and , respectively. If and intersect transversally, then is a smooth subvariety of codimension , which is denoted by . In the general case, the pair constitutes an algebraic cycle of codimension . The idea behind this definition is that and are replaced by cycles and that are equivalent in some sense but that are in general position, and one then takes the intersection of and ; of course, the cycle is also defined up to equivalence.

Let be the group of classes of algebraic cycles of codimension on modulo rational equivalence; let . Chow's intersection theory consists of the construction:

a) of a graded commutative ring structure on for each smooth quasi-projective variety ;

b) of a homomorphism of graded rings for each morphism (inverse image); and

c) of a homomorphism of groups of degree for each proper morphism (direct image).

Between the constructions a), b), c) there are a number of relations, of which the main ones are as follows:

the projection formula: For a proper morphism and cycles and ,

reduction to the diagonal: If is the diagonal morphism, and , then .

Also, there exists a natural homomorphism

which enables one to construct the theory of Chern classes (cf. Chern class) with values in Chow rings, and in particular the Chern character

which is a ring homomorphism.

It is simplest to determine the direct image homomorphism . Let be an irreducible subvariety; if , then , while if , then , where is the degree of over . By linearity, the definition is extended to cycles and classes of cycles. The inverse image homomorphism amounts to multiplication of cycles in accordance with

where is the projection and is the graph of . The definition of multiplication of cycles is given in two stages. Let, initially, and be irreducible subvarieties in that intersect properly (i.e. the codimension of is equal to the sum of the codimensions of and ). Each component of the intersection is ascribed some positive integer , which is the local multiplicity of the intersection. There are several definitions of , for example, Serre's Tor-formula:

where is the local ring , and are ideals of and , and is the length of the -module. After this, one puts

where runs through the irreducible components of .

The second stage is Chow's moving lemma: For any and on a quasi-projective variety there exists a cycle that is rationally equivalent to and that intersects properly with ; moreover, the rational equivalence class of is independent of .

The most interesting case is that of a projective variety ; applying the direct image functor to the structure morphism one obtains the mapping . In essence, the degree of a cycle is the number of points in its zero-dimensional component. The composition of multiplication with the degree enables one to measure an intersection numerically. For example, if and have complementary dimensions, then one obtains the intersection index (in algebraic geometry) (the intersection number) of and . Similarly, one obtains an intersection index for divisors :

For example, the Chow ring for the projective space is generated by the class of a hyperplane , where . Therefore, if are hypersurfaces of degrees , then (Bezout's theorem). The degree of a projective variety of dimension is defined as the intersection index of with a linear subspace of complementary dimension; if the varieties and intersect transversally, then the degree of is the product of the degrees of and .

For properly-intersecting effective divisors , but in the general case this is not true. For example, for an exceptional curve (cf. Exceptional subvariety) on a surface, .

Other theories have many of the formal properties of the theory of Chow rings: cycles modulo algebraic or numerical equivalence, -theory, singular cohomology theory (in the case ), and -adic cohomology theory (see also Weil cohomology). This leads to the axiomatic construction of intersection theory by putting each variety (from some category) into correspondence with a ring and homomorphisms and related by axioms of the type of projection or reduction-to-the-diagonal formulas (see [1]). The comparison of different intersection theories leads to useful results. For example, in the complex case, the concept of a fundamental cycle enables one to define an intersection-theory homomorphism , which enables one to use transcendental methods. Comparison of -theory with Chow's theory leads to the Riemann–Roch–Grothendieck theorem (cf. Riemann–Roch theorem). An important part is played here by the intersection theory for monoidal transformations [2], [6]. Another application of intersection theory is related to the foundations of Schubert's geometric calculus [3]. This branch of geometry can be regarded as the theory of Chow rings for various varieties that classify geometric objects: Grassmann manifolds, flag manifolds, etc.

References

[1]	, Anneaux de Chow et applications , Sem. Chevalley (1958)
[2]	Yu.I. Manin, "Lectures on algebraic geometry" , 1 , Moscow (1970) (In Russian)
[3]	"Hilbert's problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German)
[4]	M. Baldassarri, "Algebraic varieties" , Springer (1956)
[5]	J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965)
[6]	P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , Théorie des intersections et théorème de Riemann–Roch (SGA 6) , Lect. notes in math. , 225 , Springer (1971)
[7]	R. Hartshorne, "Algebraic geometry" , Springer (1977)
[8]	R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1982)
[9]	W. Fulton, "Intersection theory" , Springer (1984)

Comments

W. Fulton has defined Chow groups for singular varieties [a1]. A more refined intersection theory has been developed by Fulton and R. MacPherson: given cycles and on , it gives a well-defined element of [a2].

A new development is the intersection theory for arithmetic varieties, i.e. flat schemes over the ring of integers of number fields with suitable extra data for the infinite places, [a3]–.

References

[a1]	W. Fulton, "Rational equivalence on singular varieties" Publ. Math. IHES , 45 (1975) pp. 147–167
[a2]	W. Fulton, R.D. MacPherson, "Defining algebraic intersections" L.D. Olson (ed.) , Algebraic geometry , Lect. notes in math. , 687 , Springer (1978) pp. 1–30
[a3]	G. Faltings, "Calculus on arithmetic surfaces" Ann. of Math. , 119 (1984) pp. 387–424
[a4]	S. Arakelov, "Intersection theory of divisors on an arithmetical surface" Math. USSR Izv. , 8 (1974) pp. 1167–1180 Izv. Akad. Nauk. SSSR Ser. Mat. , 38 : 6 (1974) pp. 1179–1192
[a5]	H. Gillet, "An introduction to higher dimensional Arakelov theory" K.A. Ribet (ed.) , Current trends in arithmetical algebraic geometry , Contemp. Math. , 67 , Amer. Math. Soc. (1987) pp. 209–228