Chow coordinates

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In mathematics, and more particularly in the field of algebraic geometry, Chow coordinates are a generalization of Plücker coordinates, applying to non-linear m-dimensional algebraic varieties of degree d in Pⁿ, that is, n-dimensional projective space. They are named for W. L. Chow.

To define the Chow coordinates, take the intersection of an algebraic variety Z of degree d by linear subspaces U of appropriate codimension. When m + dim(U) = n, and 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.

[edit] References

• Hodge and Pedoe, vol. II