Chow variety

Examples of Chow varieties

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

Degree 1: Subspaces

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 a Grassmannian variety:

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

The latter space has a distinguished system of homogeneous coordinates, given by the Plücker coordinates.

Dimension 0: Points

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 a symmetric power of ${\displaystyle {\mathbb{P}}^{n-1}}$:

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

Codimension 1: Divisors

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

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

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 ${\displaystyle \mathrm{Gr}(2,2,4)}$ parametrizes dimension 1, degree 2 cycles in ${\displaystyle {\mathbb{P}}^3}$. This Chow variety has two irreducible components.

• The moduli of conics contained in a projective plane (and their degenerations).
• The moduli of pairs of lines.

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.

The Chow embedding

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 ${\displaystyle (n-k)}$-dimensional projective subspaces of ${\displaystyle {\mathbb{P}}^{n-1}}$ intersect X in d-many points. By contrast, most ${\displaystyle (n-k-1)}$-dimensional projective subspaces of ${\displaystyle {\mathbb{P}}^{n-1}}$ do not intersect at X at all. This can be sharpened as follows.

Lemma. 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 d.

As a consequence, there exists a degree d form ${\displaystyle R_X}$ 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 the Chow 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. 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 the Chow 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.

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 ${\displaystyle {\mathbb{P}}^{n-1}}$ 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.

The Hilbert scheme ${\displaystyle \mathrm{Hilb}(k,d,n)}$ is the fine moduli scheme of closed subschemes of dimension k-1 and degree d inside ${\displaystyle {\mathbb{P}}^{n-1}}$. 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 the cycle map or the Hilbert-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.

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 ${\displaystyle {\overline{M}}_{0,n}}$ of stable genus-zero curves with n marked points is the Chow quotient of Grassmannian ${\displaystyle \mathrm{Gr}(2,{\mathbb{C}}^n)}$ by the standard maximal torus.