Algebraic Geometry - Computer Graphics and Geometric Modeling - page 775

Another interesting fact about birational equivalences is that they define a bijec-
tion between places and generic points. See [Seid68].

10.13.26. Theorem. Any irreducible affine plane curve in C 2 is birationally equiva-
lent to a plane curve with only ordinary singularities.

Proof. This is only a restatement of Lemma 10.12.15 since the quadratic transfor-
mations used there are birational equivalences. See [Walk50].

If we want to end up with a plane curve, then Theorem 10.13.26 cannot be
improved. We cannot prevent ordinary singularities, that is, points where the curve
crosses itself transversally. On the other hand, if we allow ourselves to move to curves
in higher dimensions, then one can show the following:

Any affine variety in C 2

10.13.27. Theorem. is birationally equivalent to a variety

with no singularities.

Proof. See [Harr92] or [Shaf94].

Theorem 10.13.27 does not say anything about the dimension of the space that
contains the variety with no singularities. One can show that any plane curve in P 2 ( C )
is birationally equivalent to a curve in P 3 ( C ) that has no singularities ([BriK81]). See
also Theorem 10.14.7 in the next section.
Next, we discuss a class of functions defined on varieties that are especially inter-
esting, namely, the “finite” functions. They give an algebraic characterization of cov-
ering spaces. First, we need a definition.

Definition. Let S be a subring of a commutative ring R with identity and assume
that S contains that identity. An element r ΠR is said to be integral over S if

m m - 1

r + s r + ... s r + s = 0 for some s ΠS .

m - 1 1 0 i

We say that R is integral over S if every element of R is integral over S.

Let f : V Æ W be a dominant regular function between affine varieties. We know
from Proposition 10.13.4 that f*: k[ W ] Æ k[ V ] is one-to-one and so we may consider
k[ W ] as a subring of k[ V ].

Definition. We say that f is a finite map if k[ V ] is integral over k[ W ].

10.13.28. Theorem.

(1) The inverse image of every point for a finite map is a finite set.
(2) If k is algebraically closed, then a finite map is onto and takes closed sets to
closed sets.

Proof. See [Shaf94].

To be a finite map is a local property.