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) Zbl 0098.13101 |
[2] | Yu.I. Manin, "Lectures on algebraic geometry" , 1 , Moscow (1970) (In Russian) MR0284434 Zbl 0204.21302 |
[3] | "Hilbert's problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German) |
[4] | M. Baldassarri, "Algebraic varieties" , Springer (1956) MR0082172 Zbl 0995.14003 Zbl 0075.15902 |
[5] | J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965) MR0201468 Zbl 0142.28603 |
[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) MR0463157 Zbl 0367.14001 |
[8] | R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1982) |
[9] | W. Fulton, "Intersection theory" , Springer (1984) MR0735435 MR0732620 Zbl 0541.14005 |
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 MR0404257 Zbl 0332.14002 |
[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 MR0527228 Zbl 0405.14003 |
[a3] | G. Faltings, "Calculus on arithmetic surfaces" Ann. of Math. , 119 (1984) pp. 387–424 MR0740897 Zbl 0559.14005 |
[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 MR472815 |
[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 MR0902594 Zbl 0621.14004 |
Intersection theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intersection_theory&oldid=19158