Intersection theory

on an algebraic variety

The theory of intersections of algebraic subvarieties and cycles. Let $ X $ be a smooth algebraic variety of dimension $ n $ over a field $ k $, while $ Y $ and $ Z $ are subvarieties of $ X $ of codimension $ i $ and $ j $, respectively. If $ Y $ and $ Z $ intersect transversally, then $ Y \cap Z $ is a smooth subvariety of codimension $ i+ j $, which is denoted by $ Y \cdot Z $. In the general case, the pair $ ( Y, Z) $ constitutes an algebraic cycle $ Y \cdot Z $ of codimension $ i + j $. The idea behind this definition is that $ Y $ and $ Z $ are replaced by cycles $ Y ^ \prime $ and $ Z ^ \prime $ that are equivalent in some sense but that are in general position, and one then takes the intersection of $ Y ^ \prime $ and $ Z ^ \prime $; of course, the cycle $ Y ^ \prime \cdot Z ^ \prime $ is also defined up to equivalence.

Let $ A ^ {i} ( X) $ be the group of classes of algebraic cycles of codimension $ i $ on $ X $ modulo rational equivalence; let $ A( X) = \oplus _ {i \geq 0 } A ^ {i} ( X) $. Chow's intersection theory consists of the construction:

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

b) of a homomorphism of graded rings $ f ^ { \star } : A( Y) \rightarrow A( X) $ for each morphism $ f: X \rightarrow Y $( inverse image); and

c) of a homomorphism of groups $ f _ \star : A( X) \rightarrow A( Y) $ of degree $ \mathop{\rm dim} Y - \mathop{\rm dim} X $ for each proper morphism $ f: X \rightarrow Y $( 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 $ f: X \rightarrow Y $ and cycles $ \alpha \in A( X) $ and $ \beta \in A( Y) $,

$$ f _ \star ( \alpha \cdot f ^ { \star } ( \beta )) = f _ \star ( \alpha ) \cdot \beta ; $$

reduction to the diagonal: If $ \Delta : X \rightarrow X \times X $ is the diagonal morphism, and $ \alpha , \beta \in A( X) $, then $ x \cdot y = \Delta ^ \star ( \alpha \times \beta ) $.

Also, there exists a natural homomorphism

$$ c _ {1} : \mathop{\rm Pic} ( X) \rightarrow A ^ {1} ( X), $$

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

$$ \mathop{\rm ch} : K( X) \rightarrow A( X) \otimes \mathbf Q , $$

which is a ring homomorphism.

It is simplest to determine the direct image homomorphism $ f _ \star $. Let $ Z \subset X $ be an irreducible subvariety; if $ \mathop{\rm dim} f( Z) < \mathop{\rm dim} Z $, then $ f _ \star ( Z) = 0 $, while if $ \mathop{\rm dim} f( Z) = \mathop{\rm dim} Z $, then $ f _ \star ( Z) = d \cdot f ( Z) $, where $ d $ is the degree of $ Z $ over $ f( Z) $. By linearity, the definition is extended to cycles and classes of cycles. The inverse image homomorphism $ f ^ { \star } $ amounts to multiplication of cycles in accordance with

$$ f ^ { \star } ( \alpha ) = p _ \star ( \Gamma _ {f} \cdot ( X \times \alpha )), $$

where $ p: X \times Y \rightarrow X $ is the projection and $ \Gamma _ {f} \subset X \times Y $ is the graph of $ f $. The definition of multiplication of cycles is given in two stages. Let, initially, $ Y $ and $ Z $ be irreducible subvarieties in $ X $ that intersect properly (i.e. the codimension of $ Y \cap Z $ is equal to the sum of the codimensions of $ Y $ and $ Z $). Each component $ W $ of the intersection $ Y \cap Z $ is ascribed some positive integer $ i( Y, Z; W) $, which is the local multiplicity of the intersection. There are several definitions of $ i( Y, Z; W) $, for example, Serre's Tor-formula:

$$ i( Y, Z; W) = \sum _ { k\geq } 0 (- 1) ^ {k} l( \mathop{\rm Tor} _ {k} ^ {A} ( A / \mathfrak a , A / \mathfrak b )), $$

where $ A $ is the local ring $ {\mathcal O} _ {X,W} $, $ \mathfrak a $ and $ \mathfrak b $ are ideals of $ Y $ and $ Z $, and $ l $ is the length of the $ A $- module. After this, one puts

$$ Y \cdot Z = \sum _ { W } i( Y, Z; W) \cdot W, $$

where $ W $ runs through the irreducible components of $ Y \cap Z $.

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

The most interesting case is that of a projective variety $ X $; applying the direct image functor to the structure morphism $ X \rightarrow \mathop{\rm Spec} k $ one obtains the mapping $ \mathop{\rm deg} : A( X) \rightarrow \mathbf Z $. 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 $ Y $ and $ Z $ have complementary dimensions, then one obtains the intersection index (in algebraic geometry) (the intersection number) of $ Y $ and $ Z $. Similarly, one obtains an intersection index for $ n $ divisors $ D _ {1} \dots D _ {n} $:

$$ ( D _ {1} \dots D _ {n} ) = \mathop{\rm deg} ( D _ {1} \dots D _ {n} ). $$

For example, the Chow ring for the projective space $ P ^ {n} $ is generated by the class of a hyperplane $ H $, where $ ( H ^ {n} ) = ( H \dots H) = 1 $. Therefore, if $ D _ {1} \dots D _ {n} $ are hypersurfaces of degrees $ d _ {1} \dots d _ {n} $, then $ ( D _ {1} \dots D _ {n} ) = d _ {1} \dots d _ {n} $( Bezout's theorem). The degree of a projective variety $ Y \subset P ^ {n} $ of dimension $ k $ is defined as the intersection index of $ Y $ with a linear subspace $ P ^ {n-} k $ of complementary dimension; if the varieties $ Y $ and $ Z $ intersect transversally, then the degree of $ Y \cap Z $ is the product of the degrees of $ Y $ and $ Z $.

For properly-intersecting effective divisors $ ( D _ {1} \dots D _ {n} ) \geq 0 $, but in the general case this is not true. For example, for an exceptional curve (cf. Exceptional subvariety) $ E $ on a surface, $ ( E, E) = - 1 $.

Other theories have many of the formal properties of the theory of Chow rings: cycles modulo algebraic or numerical equivalence, $ K $- theory, singular cohomology theory $ H ^ \star ( , \mathbf Z ) $( in the case $ k = \mathbf C $), and $ l $- adic cohomology theory (see also Weil cohomology). This leads to the axiomatic construction of intersection theory by putting each variety $ X $( from some category) into correspondence with a ring $ C( X) $ and homomorphisms $ f ^ { \star } $ and $ f _ \star $ 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 $ A( X) \rightarrow H ^ \star ( X, \mathbf Z ) $, which enables one to use transcendental methods. Comparison of $ K $- 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 $ Y $ and $ Z $ on $ X $, it gives a well-defined element $ X \cdot Y $ of $ A ( Y \cap Z ) $[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