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In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety Gr {\displaystyle \operatorname {Gr} } is the fine moduli variety parametrizing all effective algebraic cycles of dimension k − 1 {\displaystyle k-1} and degree d {\displaystyle d} in P n − 1 {\displaystyle \mathbb {P} ^{n-1}}.
The Chow variety Gr {\displaystyle \operatorname {Gr} } may be constructed via a Chow embedding into a sufficiently large projective space. This is a direct generalization of the construction of a Grassmannian variety via the Plücker embedding, as Grassmannians are the d = 1 {\displaystyle d=1} case of Chow varieties.
Chow varieties are distinct from Chow groups, which are the abelian group of all algebraic cycles on a variety up to rational equivalence. Both are named for Wei-Liang Chow, a pioneer in the study of algebraic cycles.