Chow Coordinates

A generalization of Grassmann Coordinates to $m$-D varieties of degree $d$ in $P^n$, where $P^n$ is an $n$-D projective space. To define the Chow coordinates, take the intersection of a $m$-D Variety $Z$ of degree $d$ by an $(n-m)$-D Subspace $U$ of $P^n$. Then the coordinates of the $d$ points of intersection are algebraic functions of the Grassmann Coordinates of $U$, and by taking a symmetric function of the algebraic functions, a hHomogeneous 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.

References

Chow, W.-L. and van der Waerden., B. L. ``Zur algebraische Geometrie IX.'' Math. Ann. 113, 692-704, 1937.

Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and Igusa, J.-I. ``Wei-Liang Chow.'' Not. Amer. Math. Soc. 43, 1117-1124, 1996.