Chow coordinates
Encyclopedia
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and more particularly in the field of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, Chow coordinates are a generalization of Plücker coordinates
Plücker coordinates
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogenous coordinates to each line in projective 3-space, P3. Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines...
, applying to m-dimensional algebraic varieties of degree d in Pn, that is, n-dimensional projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
. 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
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
m. When U is in general position
General position
In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...
, 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.