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 ${\displaystyle \mathbb {P} ^{n}}$, that is, n-dimensional projective space. They are named for Wei-Liang Chow.

For the dog, see Chow 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 (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.

Relation to Hilbert scheme

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.

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

See also

• Picard variety
• GIT quotient

References

• Chow, W.-L.; van der Waerden, B. L. (1937), "Zur algebraische Geometrie IX.", Mathematische Annalen, 113: 692–704, doi:10.1007/BF01571660
• Hodge, W. V. D.; Pedoe, Daniel (1994) [1947]. Methods of Algebraic Geometry, Volume I (Book II). Cambridge University Press. ISBN 978-0-521-46900-5. MR 0028055.
• Hodge, W. V. D.; Pedoe, Daniel (1994) [1952]. Methods of Algebraic Geometry: Volume 2 Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties. Cambridge Mathematical Library. Cambridge University Press. ISBN 978-0-521-46901-2. MR 0048065.
• Mikhail Kapranov, Chow quotients of Grassmannian, I.M. Gelfand Seminar Collection, 29–110, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993.
• Kollár, János (1996), Rational Curves on Algebraic Varieties, Berlin, Heidelberg: Springer-Verlag
• Kollár, János, "Chapter 1", Book on Moduli of Surfaces
• Kulikov, Val.S. (2001) [1994], "Chow variety", Encyclopedia of Mathematics, EMS Press
• Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. 34 (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.