(Moving lemma for divisors)
Let \(X\) be a quasiprojective variety over an algebraically closed field, \(D\) a Cartier divisor on \(X\), and \(S \subseteq X\) a finite set of points. Then \(D\) is linearly equivalent to a Cartier divisor \(D'\) whose support contains no point of \(S\).
(Chow's moving lemma)
Let \(X\) be a smooth quasiprojective variety, and let \(\alpha\) and \(\beta\) be algebraic cycles on \(X\). Then \(\alpha\) is rationally equivalent to a cycle \(\alpha'\) that meets \(\beta\) properly: every irreducible component of \(\supp \alpha' \cap \supp \beta\) has codimension \(\codim \alpha + \codim \beta\).
The moving lemma is what makes the intersection product on the Chow ring \(A^*(X)\) of a smooth quasiprojective variety well defined: move one cycle to meet the other properly, intersect with multiplicities, and the rational equivalence class of the result does not depend on the choice of \(\alpha'\). For divisors on a smooth projective surface, it lets the intersection number \(C \cdot D\) be computed as a sum of local intersection multiplicities after replacing \(D\) by a linearly equivalent divisor with no common component with \(C\).
Dependencies
D-5PQ5WWeil and Cartier divisors, and when they agreeuses
Backlinks
D-CHOWRINGChow groups, the Chow ring and the cycle class mapuses