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.
is the fine moduli variety parametrizing all effective algebraic cycles of dimension
may be constructed via a Chow embedding into a sufficiently large projective space.
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.
, the degree of X is the number of intersection points between X and a generic[2]
[3] Degree is constant in families[4] of subvarieties, except in certain degenerate limits.
There are several approaches to reconciling this issue, but the simplest is to declare
-dimensional algebraic cycle is a finite formal linear combination in which
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
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.
There are three special classes of Chow varieties with particularly simple constructions.
of dimension k-1 and degree 1 is the projectivization of a k-dimensional subspace of n-dimensional affine space.
of dimension 0 and degree d is an (unordered) d-tuple of points in
of codimension 1[5] 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.
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.
These two 8-dimensional components intersect in the moduli of coplanar pairs of lines, which is the singular locus in
This shows that, in contrast with the special cases above, Chow varieties need not be smooth or irreducible.
, called the Chow form of X, which is well-defined up to scaling.
denote the vector space of degree d forms on
In particular, an effective algebraic cycle X is determined by its Chow form
in this basis gives a system of homogeneous coordinates on the Chow variety
A more sophisticated solution to the problem of 'correctly' counting the degree of a degenerate subvariety is to work with subschemes of
Schemes can keep track of infinitesimal information that varieties and algebraic cycles cannot.
is the fine moduli scheme of closed subschemes of dimension k-1 and degree d inside
[10] Each closed subscheme determines an effective algebraic cycle, and the induced map is called the cycle map or the Hilbert-Chow morphism.
corresponding to irreducible subvarieties of degree d, but the fibers over non-simple algebraic cycles can be more interesting.
A Chow quotient parametrizes closures of generic orbits.
It is constructed as a closed subvariety of a Chow variety.
of stable genus-zero curves with n marked points is the Chow quotient of Grassmannian