Algebraic Geometry - Computer Graphics and Geometric Modeling - page 782

10.13.38. Theorem. (The Noether Normalization Theorem)

(1) Any irreducible projective variety V in P n (k) admits a finite map f : V Æ P m (k)
for some m £ n.
(2) Any irreducible affine variety V in k n admits a finite map f : V Æ k m for some m £ n.

Proof. See [Shaf94]. We sketch the proof of (1). Assume that V π P n (k) and let
p ΠP n (k) - V . The projection j of V with center p will be regular. The image j( V ) in
P n-1 (k) will be a projective variety and j : V Æj( V ) is a finite map by Theorem
10.13.36. If j( V ) π P n-1 (k) then we can repeat this process. Since the composite
of a finite number of finite maps is finite we will finally get our map f.

Our coordinate rings used functions (polynomials) that were defined on the whole
variety. However, one can also get useful information from local properties. Instead
of globally defined functions one can look at local rings . These are defined for every
point and are rings of functions that are only defined in a neighborhood of the point.
See [Shaf94]. They also show up in complex analysis.
Finally, we want to draw the reader's attention to a property of the rational para-
meterization, call it j(t), of a conic that we described in the discussion after Example
10.13.1 and its bearing on the following type of problem: Given a subfield k of a field
K and a curve in K 2 defined by f(X,Y) = 0, find all the points of the curve with coor-
dinates in k. For example, we might want the rational points of the curve in Example
10.13.1. In the case of a conic, if all the coefficients of f(X,Y) and the coordinates of
the given point p 0 of the curve belonged to k, then our j(t) will generate the desired
points as t ranges over k. Other curves admit similar parameterizations.

10.14 Space Curves

In addition to plane curves, another class of spaces that have great practical interest
are space curves, specifically curves in R 3 . The simplest and intuitive definition is:

Definition. A curve or algebraic curve is an irreducible algebraic variety of dimension 1.

The only problem with this definition is that we have not yet defined what the
dimension of an algebraic variety is. We shall do so in Section 10.16, but the concept
of dimension and the associated topic of higher dimensional varieties, although very
important in algebraic geometry, is too advanced for us to do anything more than give
a brief overview. For that reason, to avoid a lengthy digression at this point, we shall
give an equivalent, but ad hoc, definition of a curve that does not use dimension and
yet will enable us to study some of their properties. Our approach, which follows that
given in [Walk50], will seem rather roundabout and a kind of “trick.” At the end of
this section we shall rephrase the definition in terms of ideals. There is one property
of dimension that the reader should be aware of right now though, otherwise it might
be puzzling why we continually restrict ourselves to transcendence degree one in the
discussion that follows. In the case of an irreducible variety, its dimension is the same
as the transcendence degree of its function field (see Theorem 10.16.9).
Note that the function field K of an affine curve in the “plane” k 2 can be expressed
in the form K = k(x,y), where x is transcendental over k and y is algebraic over k(x).