Chow variety


In mathematics, and more particularly in the field of algebraic geometry, Chow coordinates are a generalization of Plücker coordinates, applying to m-dimensional algebraic varieties of degree d in, that is, n-dimensional projective space. They are named for Wei-Liang Chow.
A Chow variety is a variety whose points correspond to all cycles of a given projective space of given dimension and degree.

Definition

To define the Chow coordinates, take the intersection of an algebraic variety Z, inside a projective space, of degree d and dimension m by linear subspaces U of codimension m. When U is in general position, the intersection will be a finite set of d distinct points.
Then the coordinates of the d points of intersection are algebraic functions of the Plücker coordinates of U, and by taking a symmetric function of the algebraic functions, a homogeneous polynomial known as the Chow form of Z is obtained.
The Chow coordinates are then the coefficients of the Chow form. Chow coordinates can generate the smallest field of definition of a divisor. The Chow coordinates define a point in the projective space corresponding to all forms.
The closure of the possible Chow coordinates is called the Chow variety.

Relation to Hilbert scheme

The Hilbert scheme is a variant of the Chow varieties. There is always a map
from the Hilbert scheme to the Chow variety.

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 of stable genus-zero curves with n marked points is the Chow quotient of Grassmannian by the standard maximal torus.