Intersection theory

Remarks

Chow's moving lemma says that such representatives $V$ and $W$ always exist, and the class of $V \cdot W$ does not depend on their choice. The product makes $CH^\bullet(X)$ a commutative, associative graded ring with unit $[X]$. For $X$ projective, the degree of a $0$-cycle gives intersection numbers $\deg(\alpha \cdot \beta)$ for $p + q = n$. Example: for $X = \mathbb{P}^n$ with $h$ the class of a hyperplane, $CH^\bullet(\mathbb{P}^n) \cong \Z[h]/(h^{n+1})$, and a subvariety of codimension $j$ and degree $d$ has class $d h^j$. Non-example: in $\mathbb{P}^2$, a line $L$ does not meet itself properly, so $L \cdot L$ is not given by the formula with $V = W = L$. One moves one copy to another line $L'$, and $[L] \cdot [L] = [L \cap L']$ is the class of a point. Fulton and MacPherson gave a construction of the product, by deformation to the normal cone, that works without the moving lemma and in greater generality.