Graphics Reference
In-Depth Information
A corresponding result holds for projective varieties in
P
n
(
C
).
Proof.
See [Kend77].
Theorem 10.18.1 leads to a definition of nonsingularity which generalizes the def-
inition we gave in the case of plane curves.
Definition.
Let
V
be an irreducible variety in
C
n
and let
p
Œ
V
. If equation (10.93)
holds at
p
, then
p
is called a
nonsingular
point of
V
and
V
is said to be
nonsingular
at
p
. Otherwise,
p
is called a
singular
point for
V
and
V
is said to be
singular at
p
. If
every point of
V
is a nonsingular point, then
V
is said to be a
nonsingular variety
. A
similar definition is made in the case of an irreducible projective variety in
P
n
(
C
).
10.18.2. Theorem.
The set of singular points of an irreducible variety in
C
n
or
P
n
(
C
)
form a proper subvariety.
Proof.
See [Kend77].
Theorems 10.18.1 and 10.18.2 generalize to arbitrary varieties. We now turn to
some topological issues.
10.18.3. Theorem.
(1) Every algebraic curve in
P
n
(
C
) is connected.
(2) Every irreducible variety in
C
n
or
P
n
(
C
) is connected.
Proof.
See [Kend77]. The reason that irreducibility is not needed in (1) is that the
irreducible components of a curve all intersect in this case and the union of connected
sets that intersect is connected.
10.18.4. Theorem.
(1) Every nonsingular one-dimensional variety in
C
n
or
P
n
(
C
) is an orientable
surface.
(2) Every irreducible d-dimensional nonsingular variety in
C
n
or
P
n
(
C
) is ori-
entable as a real 2d-dimensional manifold.
Proof.
See [Kend77]. Compare (1) with Theorem 10.2.5.
The final topic of this section deals with intersections of varieties and Bèzout's
Theorem. There will be a long list of theorems which deal with the intersection of
suitable linear subspaces with varieties. The reader should reread the comments at
the beginning of Section 10.6 for why this is a reasonable geometric approach to the
definitions and theorems we shall state. Intersections of varieties with linear sub-
spaces also played a role in Section 10.7 (in that case we used lines) and in Section
10.16 (see the comments at the end of the section regarding the definition of dimen-
sion in terms of finite maps). First, though, it is convenient to state a bound on the
dimension of an intersection in terms of its codimension.
