---
tag: 9E9T41A7XN
kind: definition
name: Intersection theory
msc: [14C17, 14C15]
defines: 4HV4WTAX9P
uses: [EQSHJPE7H8, 9VFWPZEVGN, EBEF6PEK4N, GPQF9GK4T9, 36X2WTHYCB]
sources:
  - url: https://en.wikipedia.org/wiki/Intersection_theory
---

## Definition

Let $X$ be a smooth, irreducible [[quasi-projective variety|EQSHJPE7H8]] of dimension $n$ over a field, and let $CH^j(X)$ be its [[Chow group|EBEF6PEK4N]] of codimension $j$ cycles.
Let $V$ and $W$ be irreducible subvarieties of codimensions $p$ and $q$.
They *meet properly* if every component $Z$ of $V \cap W$ has codimension $p + q$.
In that case set $V \cdot W = \sum_Z i(Z; V \cdot W) \, [Z]$, where $i(Z; V \cdot W)$ is the [[intersection multiplicity|GPQF9GK4T9]].
Extend this bilinearly to [[cycles|9VFWPZEVGN]] whose components meet properly.
The *intersection product* is the map $CH^p(X) \times CH^q(X) \to CH^{p+q}(X)$, $(\alpha, \beta) \mapsto \alpha \cdot \beta$, defined as follows.
Choose cycles $V$ in $\alpha$ and $W$ in $\beta$ that meet properly, and let $\alpha \cdot \beta$ be the class of $V \cdot W$.
With this product, $CH^\bullet(X) = \bigoplus_{j=0}^n CH^j(X)$ is the *Chow ring* of $X$.
*Intersection theory* is the study of this ring and of the maps between such rings.

## 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.