algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, Chow's moving lemma, proved by , states: given algebraic cycles ''Y'', ''Z'' on a nonsingular

quasi-projective variety In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski topology, Zariski-closed subset. A similar defin ...

''X'', there is another algebraic cycle ''Z' '' which is rationally equivalent to ''Z'' on ''X,'' such that ''Y'' and ''Z' '' intersect properly. The lemma is one of the key ingredients in developing

intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...

and the

Chow ring In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so ...

, as it is used to show the uniqueness of the theory. Even if ''Z'' is an effective cycle, it is not, in general, possible to choose ''Z' '' to be effective.

References

* * * Theorems in algebraic geometry Zhou, Weiliang {{math-hist-stub