Graphics Reference
In-Depth Information
of
V
. The (
complex
)
dimension
of
V
at
such
a smooth point
is then defined to be n -
r and is denoted by dim
p
(
V
). Similar definitions are made for varieties in
P
n
(
C
). A
variety is called a
smooth variety
if every one of its points is a smooth point.
Note that the dimension in the definition is a
complex
dimension, so that as a
real manifold, the variety would have
real
dimension 2dim
p
(
V
) at
p
.
The set of smooth points of a variety in either
C
n
or
P
n
(
C
) are
10.16.5. Theorem.
dense in the variety.
Proof.
See [Kend77].
Theorem 10.16.5 implies that the next definition is well defined.
Definition.
The
dimension
of a variety
V
at a point
p
in
V
, denoted by dim
p
(
V
), is
defined by
()
=
{
()
}
dim
V
lim max dim
V
q
is a smooth point in
U
,
p
q
i
U
i
where
U
i
is any sequence of neighborhoods of
p
that converge to
p
. In other words,
dim
p
(
V
) is the maximum of the dimensions of
V
at smooth points in an arbitrarily
small neighborhood of
p
. The
dimension
of
V
, denoted by dim
V
, is defined by
dim
V
=-
1
,
if
V
is empty,
{
()
}
= max dim
VpV
Œ
,
otherwise.
p
The
codimension
of
V
, denoted by codim
V
, is defined by
codim
V
=-
n
dim
V
.
In an irreducible variety in
C
n
or
P
n
(
C
) all points have the same
10.16.6. Theorem.
dimension.
Proof.
See [Kend77].
Definition.
A variety is said to have
pure dimension
d if it has the same dimension
d at each of its points.
Every irreducible variety in
C
n
or
P
n
(
C
) has pure dimension.
10.16.7. Corollary.
A variety in
C
n
or
P
n
(
C
) is a hypersurface if and only if it has
10.16.8. Theorem.
pure dimension n - 1.
Proof.
See [Kend77].
In the next section we shall carry this definition of dimension further to study sin-
gularities, intersection multiplicities, and other concepts that we dealt with in the case
