# 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

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].