Inmathematics, particularly in the field ofalgebraic geometry, aChow variety is analgebraic variety whose points correspond to effectivealgebraic cycles of fixed dimension and degree on a givenprojective space. More precisely, the Chow variety^[1]${\displaystyle \mathrm{Gr}(k,d,n)}$ is thefine moduli variety parametrizing all effective algebraic cycles of dimension${\displaystyle k-1}$ and degree${\displaystyle d}$ in${\displaystyle {\mathbb{P}}^{n-1}}$.

The Chow variety${\displaystyle \mathrm{Gr}(k,d,n)}$ may be constructed via aChow embedding into a sufficiently large projective space. This is a direct generalization of the construction of aGrassmannian variety via thePlücker embedding, as Grassmannians are the${\displaystyle d=1}$ case of Chow varieties.

Chow varieties are distinct fromChow groups, which are the abelian group of allalgebraic cycles on a variety (not necessarily projective space) up to rational equivalence. Both are named forWei-Liang Chow (周煒良), a pioneer in the study of algebraic cycles.

If X is a closedsubvariety of${\displaystyle {\mathbb{P}}^{n-1}}$ of dimension${\displaystyle k-1}$, thedegree of X is the number of intersection points between X and a generic^[2]${\displaystyle (n-k)}$-dimensionalprojective subspace of${\displaystyle {\mathbb{P}}^{n-1}}$.^[3]

Degree is constant in families^[4] of subvarieties, except in certain degenerate limits. To see this, consider the following family parametrized by t.

${\displaystyle X_t:=V(x^2-tyz)\subset {\mathbb{P}}^2}$.

Whenever${\displaystyle t\ne 0}$,${\displaystyle X_t}$ is a conic (an irreducible subvariety of degree 2), but${\displaystyle X_0}$ degenerates to the line${\displaystyle x=0}$ (which has degree 1). There are several approaches to reconciling this issue, but the simplest is to declare${\displaystyle X_0}$ to be aline of multiplicity 2 (and more generally to attach multiplicities to subvarieties) using the language ofalgebraic cycles.

A${\displaystyle (k-1)}$-dimensionalalgebraic cycle is a finite formal linear combination

${\displaystyle X=\sum _i{m}_i{X}_i}$.

in which${\displaystyle X_i}$s are${\displaystyle (k-1)}$-dimensional irreducible closed subvarieties in${\displaystyle {\mathbb{P}}^{n-1}}$, and${\displaystyle m_i}$s are integers. An algebraic cycle iseffective if each${\displaystyle m_i\ge 0}$. Thedegree of an algebraic cycle is defined to be

${\displaystyle \mathrm{deg}(X):=\sum _i{m}_i\mathrm{deg}(X_i)}$.

Ahomogeneous polynomial orhomogeneous ideal in n-many variables defines an effective algebraic cycle in${\displaystyle {\mathbb{P}}^{n-1}}$, in which the multiplicity of each irreducible component is the order of vanishing at that component. In the family of algebraic cycles defined by${\displaystyle x^2-tyz}$, the${\displaystyle t=0}$ cycle is 2 times the line${\displaystyle x=0}$, which has degree 2. More generally, the degree of an algebraic cycle is constant in families, and so it makes sense to consider themoduli problem of effective algebraic cycles of fixed dimension and degree.

There are three special classes of Chow varieties with particularly simple constructions.

An effective algebraic cycle in${\displaystyle {\mathbb{P}}^{n-1}}$ 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 aGrassmannian variety:

${\displaystyle \mathrm{Gr}(k,1,n)\simeq \mathrm{Gr}(k,n)}$

The latter space has a distinguished system ofhomogeneous coordinates, given by thePlücker coordinates.

An effective algebraic cycle in${\displaystyle {\mathbb{P}}^{n-1}}$ of dimension 0 and degree d is an (unordered) d-tuple of points in${\displaystyle {\mathbb{P}}^{n-1}}$, possibly with repetition. This gives an isomorphism to asymmetric power of${\displaystyle {\mathbb{P}}^{n-1}}$:

${\displaystyle \mathrm{Gr}(1,d,n)\simeq {\mathrm{Sym}}_d{\mathbb{P}}^{n-1}}$.

An effective algebraic cycle in${\displaystyle {\mathbb{P}}^{n-1}}$ 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. Letting${\displaystyle V_{d,n}}$ denote the vector space of degree d polynomials in n-many variables, this gives an isomorphism to aprojective space:

${\displaystyle \mathrm{Gr}(n-1,d,n)\simeq \mathbb{P}{V}_{d,n}}$.

Note that the latter space has a distinguished system ofhomogeneous coordinates, which send a polynomial to the coefficient of a fixed monomial.

The Chow variety${\displaystyle \mathrm{Gr}(2,2,4)}$ parametrizes dimension 1, degree 2 cycles in${\displaystyle {\mathbb{P}}^3}$. 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${\displaystyle \mathrm{Gr}(2,2,4)}$. This shows that, in contrast with the special cases above, Chow varieties need not be smooth or irreducible.

Let X be an irreducible subvariety in${\displaystyle {\mathbb{P}}^{n-1}}$ of dimension k-1 and degree d. By the definition of the degree, most${\displaystyle (n-k)}$-dimensionalprojective subspaces of${\displaystyle {\mathbb{P}}^{n-1}}$ intersect X in d-many points. By contrast, most${\displaystyle (n-k-1)}$-dimensionalprojective subspaces of${\displaystyle {\mathbb{P}}^{n-1}}$ do not intersect at X at all. This can be sharpened as follows.

Lemma.^[6] The set${\displaystyle Z(X)\subset \mathrm{Gr}(n-k,n)}$ parametrizing the subspaces of${\displaystyle {\mathbb{P}}^{n-1}}$ which intersect X non-trivially is an irreducible hypersurface of degree^[7] d.

As a consequence, there exists a degree d form^[8]${\displaystyle R_X}$ on${\displaystyle \mathrm{Gr}(n-k,n)}$ which vanishes precisely on${\displaystyle Z(X)}$, and this form is unique up to scaling. This construction can be extended to an algebraic cycle${\displaystyle X=\sum _i{m}_i{X}_i}$ by declaring that${\displaystyle R_X:=\prod _i{R}_{X_i}^{m_i}}$. To each degree d algebraic cycle, this associates a degree d form${\displaystyle R_X}$ on${\displaystyle \mathrm{Gr}(n-k,n)}$, called theChow form of X, which is well-defined up to scaling.

Let${\displaystyle V_{k,d,n}}$ denote the vector space of degree d forms on${\displaystyle \mathrm{Gr}(n-k,n)}$.

The Chow-van-der-Waerden Theorem.^[9] The map${\displaystyle \mathrm{Gr}(k,d,n)\hookrightarrow \mathbb{P}{V}_{k,d,n}}$ which sends${\displaystyle X\mapsto R_X}$ is a closed embedding of varieties.

In particular, an effective algebraic cycle X is determined by its Chow form${\displaystyle R_X}$.

If a basis for${\displaystyle V_{k,d,n}}$ has been chosen, sending${\displaystyle X}$ to the coefficients of${\displaystyle R_X}$ in this basis gives a system of homogeneous coordinates on the Chow variety${\displaystyle \mathrm{Gr}(k,d,n)}$, called theChow coordinates of${\displaystyle X}$. However, as there is no consensus as to the ‘best’ basis for${\displaystyle V_{k,d,n}}$, this term can be ambiguous.

From a foundational perspective, the above theorem is usually used as the definition of${\displaystyle \mathrm{Gr}(k,d,n)}$. That is, the Chow variety is usually defined as a subvariety of${\displaystyle \mathbb{P}{V}_{k,d,n}}$, and only then shown to be a fine moduli space for the moduli problem in question.

A more sophisticated solution to the problem of 'correctly' counting the degree of a degenerate subvariety is to work withsubschemes of${\displaystyle {\mathbb{P}}^{n-1}}$ 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.

TheHilbert scheme${\displaystyle \mathrm{Hilb}(k,d,n)}$ is thefine moduli scheme of closed subschemes of dimension k-1 and degree d inside${\displaystyle {\mathbb{P}}^{n-1}}$.^[10] Each closed subscheme determines an effective algebraic cycle, and the induced map

${\displaystyle \mathrm{Hilb}(k,d,n)⟶\mathrm{Gr}(k,d,n)}$.

is called thecycle map or theHilbert-Chow morphism. This map is generically an isomorphism over the points in${\displaystyle \mathrm{Gr}(k,d,n)}$ corresponding to irreducible subvarieties of degree d, but the fibers over non-simple algebraic cycles can be more interesting.

AChow quotient parametrizes closures ofgeneric orbits. It is constructed as a closed subvariety of a Chow variety.

Kapranov's theorem says that themoduli space${\displaystyle {\overline{M}}_{0,n}}$ ofstable genus-zero curves withn marked points is the Chow quotient of Grassmannian${\displaystyle \mathrm{Gr}(2,{\mathbb{C}}^n)}$ by the standard maximal torus.

• Picard variety
• GIT quotient

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• Algebraic geometry