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 formula_1 is the fine moduli variety parametrizing all effective algebraic cycles of dimension formula_2 and degree formula_3 in formula_4.
The Chow variety formula_1 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 formula_6 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.
Background on algebraic cycles.
If X is a closed subvariety of formula_4 of dimension formula_8, the degree of X is the number of intersection points between X and a generic formula_9-dimensional projective subspace of formula_4.
Degree is constant in families of subvarieties, except in certain degenerate limits. To see this, consider the following family parametrized by t.
Whenever formula_12, formula_13 is a conic (an irreducible subvariety of degree 2), but formula_14 degenerates to the line formula_15 (which has degree 1). There are several approaches to reconciling this issue, but the simplest is to declare formula_14 to be a "line of multiplicity 2" (and more generally to attach multiplicities to subvarieties) using the language of "algebraic cycles".
A formula_2-dimensional algebraic cycle is a finite formal linear combination
in which formula_19s are formula_2-dimensional irreducible closed subvarieties in formula_4, and formula_22s are integers. An algebraic cycle is effective if each formula_23. 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 formula_4, in which the multiplicity of each irreducible component is the order of vanishing at that component. In the family of algebraic cycles defined by formula_26, the formula_27 cycle is 2 times the line formula_15, which has degree 2. More generally, the degree of an algebraic cycle is constant in families, and so it makes sense to consider the moduli problem of effective algebraic cycles of fixed dimension and degree.
Examples of Chow varieties.
There are three special classes of Chow varieties with particularly simple constructions.
Degree 1: Subspaces.
An effective algebraic cycle in formula_4 of dimension k-1 and degree 1 is the projectivization of a k-dimensional subspace of n-dimensional affine space. This gives an isomorphism to a Grassmannian variety:
The latter space has a distinguished system of homogeneous coordinates, given by the Plücker coordinates.
Dimension 0: Points.
An effective algebraic cycle in formula_4 of dimension 0 and degree d is an (unordered) d-tuple of points in formula_4, possibly with repetition. This gives an isomorphism to a symmetric power of formula_4:
Codimension 1: Divisors.
An effective algebraic cycle in formula_4 of codimension 1 and degree d can be defined by the vanishing of a single degree d polynomial in n-many variables, and this polynomial is unique up to rescaling. Letting formula_36 denote the vector space of degree d polynomials in n-many variables, this gives an isomorphism to a projective space:
Note that the latter space has a distinguished system of homogeneous coordinates, which send a polynomial to the coefficient of a fixed monomial.
A non-trivial example.
The Chow variety formula_38 parametrizes dimension 1, degree 2 cycles in formula_39. This Chow variety has two irreducible components.
These two 8-dimensional components intersect in the moduli of coplanar pairs of lines, which is the singular locus in formula_38. This shows that, in contrast with the special cases above, Chow varieties need not be smooth or irreducible. 
The Chow embedding.
Let X be an irreducible subvariety in formula_4 of dimension k-1 and degree d. By the definition of the degree, most formula_9-dimensional projective subspaces of formula_4 intersect X in d-many points. By contrast, most formula_44-dimensional projective subspaces of formula_4 do not intersect at X at all. This can be sharpened as follows.
Lemma. The set formula_46 parametrizing the subspaces of formula_4 which intersect X non-trivially is an irreducible hypersurface of degree d.
As a consequence, there exists a degree d form formula_48 on formula_49 which vanishes precisely on formula_50, and this form is unique up to scaling. This construction can be extended to an algebraic cycle formula_18 by declaring that formula_52. To each degree d algebraic cycle, this associates a degree d form formula_48 on formula_49, called the Chow form of X, which is well-defined up to scaling.
Let formula_55 denote the vector space of degree d forms on formula_49.
The Chow-van-der-Waerden Theorem. The map formula_57 which sends formula_58 is a closed embedding of varieties.
In particular, an effective algebraic cycle X is determined by its Chow form formula_48. 
If a basis for formula_55 has been chosen, sending formula_61 to the coefficients of formula_48 in this basis gives a system of homogeneous coordinates on the Chow variety formula_1, called the Chow coordinates of formula_61. However, as there is no consensus as to the ‘best’ basis for formula_55, this term can be ambiguous.
From a foundational perspective, the above theorem is usually used as the definition of formula_1. That is, the Chow variety is usually defined as a subvariety of formula_67, and only then shown to be a fine moduli space for the moduli problem in question.
Relation to the Hilbert scheme.
A more sophisticated solution to the problem of 'correctly' counting the degree of a degenerate subvariety is to work with subschemes of formula_4 rather than subvarieties. Schemes can keep track of infinitesimal information that varieties and algebraic cycles cannot.
For example, if two points in a variety approach each other in an algebraic family, the limiting subvariety is a single point, the limiting algebraic cycle is a point with multiplicity 2, and the limiting subscheme is a 'fat point' which contains the tangent direction along which the two points collided.
The Hilbert scheme formula_69 is the fine moduli scheme of closed subschemes of dimension k-1 and degree d inside formula_4. Each closed subscheme determines an effective algebraic cycle, and the induced map
is called the cycle map or the Hilbert-Chow morphism. This map is generically an isomorphism over the points in formula_1 corresponding to irreducible subvarieties of degree d, but the fibers over non-simple algebraic cycles can be more interesting.
Chow quotient.
A Chow quotient parametrizes closures of generic orbits. It is constructed as a closed subvariety of a Chow variety.
Kapranov's theorem says that the moduli space formula_73 of stable genus-zero curves with "n" marked points is the Chow quotient of Grassmannian formula_74 by the standard maximal torus.