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In-Depth Information
{
()
π
}
pV p
Œ
f
0 ,
where f is a polynomial in k[X
1
,X
2
,...,X
n
] that is homogeneous if
V
is a projective
variety, define a base for the open sets of the Zariski topology.
The Zariski topology is a topology because arbitrary intersections of varieties are
varieties by Lemma 10.8.2(2). This new topology on a variety is quite different from
the topology induced by the standard topology of k
n
or
P
n
(k) when k =
R
or
C
. For
one thing, all open sets intersect. (See Exercise 10.8.6 for some of its properties.)
Nevertheless, it is convenient terminology which is often used in algebraic geometry.
Varieties are sometimes called
closed sets
and some authors call an open subset of a
projective variety a
quasiprojective variety
. The latter term is in an attempt to unify
the concept of affine and projective variety. Projective varieties are clearly quasipro-
jective varieties, but affine varieties over
C
are also because of Theorem 10.3.6 on the
projective completion of an affine variety. On the other hand, it turns out that the set
of quasiprojective varieties is bigger than the set of affine and projective varieties. A
lot of what we do in this chapter could be done in terms of quasiprojective varieties,
but we shall not in order to cut down on the abstraction. At any rate, we shall run
into the Zariski topology again at several places later on. From now on any topolog-
ical statements about varieties will refer to it unless otherwise stated.
We finish this section by describing the algebraic analog of the projective
completion.
Definition.
If I is an ideal in k[X
1
,X
2
,...,X
n
], then
()
=
{
()
}
HI
Hf
f
Œ
I
is called the
homogenization of I
.
10.8.21. Proposition.
If I is an ideal in k[X
1
,X
2
,...,X
n
], then, H(I) is a homoge-
neous ideal in k[X
1
,X
2
,...,X
n+1
].
Proof.
Easy.
Definition.
Let
V
be an (affine) variety in k
n
. The projective variety V(H(I(
V
))) in
P
n
(k) is called the
projective closure
of
V
.
Let
V
be an affine variety in k
n
.
10.8.22. Theorem.
(1) The projective closure of
V
is the same as the projective completion of
V
.
(2) If
V
is irreducible, then so is the projective closure of
V
.
Proof.
The proof is not hard. See [CoLO97].
10.9
Defining Parameterized Curves Implicitly
Suppose that we have a curve in the plane whose points are parameterized by rational
functions, say,
