Chow coordinates

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 Pⁿ, that is, n-dimensional projective space. They are named for W. L. Chow. A Chow variety is a variety whose points correspond to all cycles of a given projective space of given dimension and degree.

To define the Chow coordinates, take the intersection of an algebraic variety Z 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, and the closure of the possible Chow coordinates is called the Chow variety.

See also

• Hilbert scheme, a sort of refinement of the Chow varieties.

References

• Chow, W.-L.; van der Waerden., B. L. (1937), "Zur algebraische Geometrie IX.", Math. Ann. 113: 692–704, doi:10.1007/BF01571660 
• Hodge and Pedoe, vol. II
• Kulikov, Val.S. (2001), "Chow variety", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/C/c022180.htm