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  "body": "\n## Definition\n\nLet $X$ 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.\nLet $V$ and $W$ be irreducible subvarieties of codimensions $p$ and $q$.\nThey *meet properly* if every component $Z$ of $V \\cap W$ has codimension $p + q$.\nIn 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]].\nExtend this bilinearly to [[cycles|9VFWPZEVGN]] whose components meet properly.\nThe *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.\nChoose cycles $V$ in $\\alpha$ and $W$ in $\\beta$ that meet properly, and let $\\alpha \\cdot \\beta$ be the class of $V \\cdot W$.\nWith this product, $CH^\\bullet(X) = \\bigoplus_{j=0}^n CH^j(X)$ is the *Chow ring* of $X$.\n*Intersection theory* is the study of this ring and of the maps between such rings.\n\n## Remarks\n\nChow'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.\nThe product makes $CH^\\bullet(X)$ a commutative, associative graded ring with unit $[X]$.\nFor $X$ projective, the degree of a $0$-cycle gives intersection numbers $\\deg(\\alpha \\cdot \\beta)$ for $p + q = n$.\nExample: 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 $d h^j$.\nNon-example: in $\\mathbb{P}^2$, a line $L$ does not meet itself properly, so $L \\cdot L$ is not given by the formula with $V = W = L$.\nOne moves one copy to another line $L'$, and $[L] \\cdot [L] = [L \\cap L']$ is the class of a point.\nFulton and MacPherson gave a construction of the product, by deformation to the normal cone, that works without the moving lemma and in greater generality.\n",
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