Chow variety

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 (or Cayley 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.

The Hilbert scheme is a variant of the Chow varieties. There is always a map (called the cycle map)

${\displaystyle \mathbf {Hilb} \to \mathbf {Cycl} ,\,Z\mapsto [Z]}$

from the Hilbert scheme to the Chow variety.

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 ${\displaystyle \operatorname {Gr} (2,\mathbb {C} ^{n})}$ by the standard maximal torus.

• Picard variety
• GIT quotient

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