Algebraic Geometry - Computer Graphics and Geometric Modeling - page 804

complex variety of P n ( C ). If V is a complex manifold of dimension 2k, then V is
compact and orientable and a fundamental homology class of V determines a homol-
ogy class in H 2k ( P n ( C )), which we shall denote by [ V ]. In fact, an arbitrary projective
variety determines such a class [ V ] not just one that has no singularities. But the topol-
ogy of P n ( C ) is well-understood and it is known that H 2k ( P n ( C )) is isomorphic to Z .
Therefore, if s be a generator, then there is a d so that [ V ] = ds. One can show that
deg V = |d|. Finally, there is a close connection between the degree of intersection of
two varieties and their topological intersection numbers, with the final result that
Bèzout's theorem can be proved via algebraic topology. We refer reader to [BriK81]
and [Harr92].
Finite maps are almost imbeddings. This leads to the question as to whether we
can find finite maps which are imbeddings and how large the n has to be. The next
theorem is the analog of the Whitney imbedding theorem for differentiable manifolds
in the algebraic setting (see the comments following Theorem 8.8.7).

10.18.18. Theorem. A nonsingular projective n-dimensional variety is isomorphic
to a subvariety of P 2n+1 .

Proof. See [Shaf94].

Every nonsingular “quasi-projective” curve is isomorphic to a curve in P 3 . Recall
from the discussion after Theorem 10.14.7 that not every curve is isomorphic to a
nonsingular one in P 2 .
The last result of this section returns to the subject of resolution of singularities
and blowups. We looked at an aspect of this for curves in Section 10.12. Blowups are
basically a special type of birational map. They are important in the study of rational
maps. Lack of space prevents us from going into any details here. The reader is
referred to [Harr92] and [Shaf94]. [Harr92] also discusses the following theorem of
H. Hironaka (his proof assumes a field of characteristic zero with the general case not
yet known):

10.18.19. Theorem. For any variety V we can find a smooth variety W and a regular
birational map j : W Æ V . The map j is called a resolution of the singularities of V .

This brief overview of algebraic geometry in higher dimensions consisted mainly
of a collection of definitions and unproved theorems but hopefully it gave the reader
at least a slight idea of the subject. If the reader is left with the feeling excessive
abstractness, of theorems that were true because the definitions were formulated in
such a way as to make them true, then this is quite understandable. This is not the
only place in mathematics where one encounters such a phenomenon. It definitely
does not mean that it is all abstract nonsense though, because one gets concrete results
at the end. As we have said before, it is coming up with the right definitions that pick
out the essential aspects of a problem that often leads to a breakthrough in the subject
and makes everything seem simple to prove afterwards.
In closing we should mention the following. We have studied varieties as subsets
of k n or P n (k), k = R or C , but one can also study them intrinsically the way one studies
abstract manifolds. They can be given intrinsic differentiable and analytic structures
and one can do calculus on them.