Chow variety

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 is the fine moduli variety parametrizing all effective algebraic cycles of dimension and degree in . The Chow variety 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 case of Chow varieties. Chow varieties are distinct from Chow groups, which are the abelian group of all algebraic cycles on a variety (not necessarily projective space) up to rational equivalence. Both are named for Wei-Liang Chow(周煒良), a pioneer in the study of algebraic cycles. If X is a closed subvariety of of dimension , the degree of X is the number of intersection points between X and a generic -dimensional projective subspace of . Degree is constant in families of subvarieties, except in certain degenerate limits. To see this, consider the following family parametrized by t. Whenever , is a conic (an irreducible subvariety of degree 2), but degenerates to the line (which has degree 1). There are several approaches to reconciling this issue, but the simplest is to declare to be a line of multiplicity 2 (and more generally to attach multiplicities to subvarieties) using the language of algebraic cycles. A -dimensional algebraic cycle is a finite formal linear combination in which s are -dimensional irreducible closed subvarieties in , and s are integers. An algebraic cycle is effective if each . The degree of an algebraic cycle is defined to be A homogeneous polynomial or homogeneous ideal in n-many variables defines an effective algebraic cycle in , in which the multiplicity of each irreducible component is the order of vanishing at that component. In the family of algebraic cycles defined by , the cycle is 2 times the line , which has degree 2.