Algebraic Geometry - Computer Graphics and Geometric Modeling - page 776

10.13.29. Theorem. Let f : V Æ W be a regular map between affine varieties. If every
point of W has a neighborhood so that the inverse image under f of every point in
that neighborhood is a finite set, then f is finite.

Proof. See [Shaf94].

Up to now we have discussed functions defined on affine varieties. It is possible
to define regular and rational functions for projective varieties, but things get more
complicated. Recall how we restricted ourselves to homogeneous polynomials in order
to get a well-defined definition for a projective variety. We need to make similar restric-
tions here.

Definition. A rational homogeneous polynomial in n + 1 variables X 1 , X 2 ,..., X n+1
is a rational polynomial function of the form

( ) fX X , ,..., X

12 n

n

+
+

1

( ) =

hX X , ,..., X ,

12 n + 1

( )

gX X , ,..., X

12 1

where f and g are homogeneous polynomials of the same degree. Let V = V(g) be the
set of points where the denominator g vanishes. The function

n

() -

P k V Æ k

[ ] Æ

( )

XX , ,..., X hXX , ,..., X

12 n + 1 12 n + 1

is called a rational function on P n (k) and will also be denoted by h or f/g.

It is easy to check that a rational homogeneous polynomial in n + 1 variables
defines a well-defined rational function on P n (k).

Definition. Let h = f/g be a rational homogeneous polynomial in n + 1 variables. If
p Œ P n (k) and g( p ) π 0, then the rational function h on P n (k) is said to be regular at
p . If h is regular at all points of a set X Õ P n (k), then it is called a regular function on
X . The set of regular functions on X is denoted by k[ X ].

Clearly, the natural addition and multiplication make k[ X ] into a ring.

kX X

[

, ,..., X

] 12 n + 1

[] ª

10.13.30. Proposition. k X .

() I X

Proof. See [Harr92].

It is also easy to show that if we restrict a regular function on a projective variety
to some affine part with respect to some parameterization, then the new notion of
regular agrees with the same notion for affine varieties. There is one big difference
however. The only regular functions on irreducible projective varieties are the con-
stants (Theorem 10.13.32). This does not happen in the affine case.

Definition. Let f : V Æ W be a map between projective varieties with W Õ P n (k). Let
p ΠV . Assume that the following two properties hold: