1.3 Results
2.1.1 Chow Groups and Chow Rings
When working with Chow groups and Chow rings by variety we will mean a re- duced and irreducible scheme. A subvariety of a scheme will be taken to mean a reduced and irreducible subscheme.
Let Y be a scheme of finite type over a ground field, (for example Y could be a variety), we may define the group of cycles onY, Z(Y), as the free abelian group generated by set of irreducible subvarieties ofY. This group is graded by dimension withZj(Y) denoting the group of j-cycles, that is the group of cycles which are finite formal linear combinations of varieties of dimension j, we can write this as
Zj(Y)= X i ni[Vi]|ni ∈Z, Vi is a jdimensional subvariety ofY .
So we have that Z(Y)= dimY M j Zj(Y). Chow Groups
The Chow group will be given by the group of cycles modulo rational equivalence. Informally we say two cycles α, β ∈ Z(Y) are rationally equivalent if there exists a “family” of cycles specified by a rational parametrization which interpolates be- tweenα andβ. More explicitly we define a map δY : Z(Y ×P1) → Z(Y) on free generators as follows. LetW be a subvariety ofY×P1. If the projection onto the second factor π : W → P1 is not dominant, i.e. ifW ⊂ Y × {t} for somet ∈ P1, then we setδY(W) = 0. If, on the other hand, the projectionW → P1 is dominant then we letW0 = π−1(0) ⊂ Y × {0} = Y andW∞ = π−1(∞) ⊂ Y× {∞} = Y, where
0= (0 : 1) and∞= (1 : 0) are the usual zero and infinity points ofP1. In this case we defineδY(W)=[W0]−[W∞].
We write Rat(Y)⊂ Z(Y) for the imageδY(Z(Y×P1)), that is the subgoup generated by all cycles of the form [W0]−[W∞]. Two cycles α, β ∈ Z(Y) are defined to be
rationally equivalent ifα−β ∈ Rat(Y), and Ratj(Y) is the group of j-cycles ratio- nally equivalent to zero. The Chow groupA(Y) is the group of rational equivalence classes,
A(Y) :=Z(Y)/Rat(Y) =coker (δY).
See Chapter 4 of Eisenbid and Harris [7] and Section 1.6 of Fulton [10] for further details.
The quotient group Aj(X) = Zj(X)/Bj(X) is the Chow group of dimensionk. The quotient group Aj(X) = Zj(X)/Bj(X) is the Chow group of codimension j where
Zj(X) is the group of codimension j cycles and Bj(X) is the associated group of codimension j cycles rationally equivalent to zero. See Chapter 1 of Fulton [10],
Chapter 9 of Gathmann [11], or §1.1 of Eisenbud and Harris [7] for a complete description.
Chow Rings
In the case where Y is a smooth variety of dimension n the Chow groups of Y,
Aj(Y), form a graded ring
A∗(Y)=
n
M
j=0
Aj(Y), (2.1)
this ring is graded by dimension. We may also form a ring A∗(Y) = A
∗(Y) graded
by codimenion from the Chow groups, that is
A∗(Y)= n
M
j=0
Aj(Y). (2.2)
Multiplication on the Chow ringA∗(Y)= A
∗(Y) is given by the intersection product
(2.3), we describe this multiplication below.
Let V be a subscheme of Y having pure codimension d, and let W be a purely
j dimensional subscheme of Y. Also let T denote the tangent bundle of Y, TY, restricted toV ∩W, and letc(T) denote the total Chern class of the vector bundle
T. We may define the intersection product as
[V]·[W]= {c(T)· s(V∩W,V ×W)}j−d ∈Aj−d(V∩W)⊂ Aj−d(Y). (2.3)
Here we considerV∩W as a subvariety ofV×W via the diagonal embedding ofY
inY×Y. Note that the expressionc(T)·s(V∩W,V×W) denotes the homomorphism specified by the Chern classc(T) acting ons(V∩W,V×W) in the manner of Fulton [10, Chapter 3]. This product makesA∗(Y) (andA∗(Y)) into a commutative graded
ring with unit [Y]. In what follows we will most frequently use the notationA∗(Y) for the Chow ring, i.e. we will use the codimension grading.
Example 2.1.1(Ex. 8.1.11 [10]). Let V,W be subvarieties of a non-singular variety Y. If V and W are non-singular varieties which meet transversely at generic points
of V ∩W we have
[V]·[W]= [V∩W]. (2.4)
More generally the equality (2.4) holds if the diagonal embedding of the intersection scheme V∩W in V ×W is a regular embedding of codimensiondimY.
Degree of a Chow Ring Element
TakeMto be a smooth (irriducible) variety over an algebraically closed field and let α= P
VnV[V] be an arbitary element of A∗(M). We will refer to
R
αas the degree of the zero dimensional part ofα, that is
Z α= X [V]∈A0(M) nV = X dim(V)=0 nV. (2.5)
Put another way, R α denotes the sum of the integer coefficients of the classes of dimension zero irreducible varieties inα, that is the coefficients of the pieces ofα which are in the dimension zero Chow groupA0(M).
Chow Ring ofPn
In this chapter (and in Chapter3) we will work only in the Chow ring ofPn,A∗(Pn) Z[h]/(hn+1), whereh = c1(OPn(1)) is the equivalence class of a hyperplane in P
n , hence a hypersurfaceWof degreedinPnis represented as [W]= d·hinA∗(Pn) (for more details see Fulton [10]). Herec1denotes the first Chern class of a line bundle, see Fulton [10,§2.5].
For an element α ∈ A∗(Pn) we have that
R
α will be the integer coefficient of hn
in α(which can be zero). For V a subscheme of pure dimension j in Pn we will write deg([V])= Z c1(OPn(1)) j [V]= Z hj[V],
that is deg([V]) is the coefficient ς ∈ Z of hn−j in [V], note that the term ςhn−j is
in the dimension j Chow group Aj(Pn). Also note that deg([V]) = deg(V) where deg(V) is the usual geometric notion of degree, i.e. deg(V) denotes the number of points in the intersection ofV with jgeneral hyperplanes.
Finally we note that in practice we will always use the presentation A∗(Pn) Z[h]/(hn+1) and hence the Segre class and cS M class will be represented as a poly- nomial with integer coefficients inZ[h]/(hn+1).