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where f
i
is the homogeneous components of degree i for f, then f vanishes at a
point
p
Œ
P
n
(k) if and only if each f
i
vanishes at
p
. (
Hint:
For fixed (c
1
,c
2
,...,c
n+1
)
consider
()
=
(
)
=+ + +
d
gt
ftc tc
,
,...,
tc
f
tf
...
t f
.)
12
n
+
1
0 1
d
(b)
Let S be a set of polynomials in k[X
1
,X
2
,...,X
n+1
]. Prove that there are a finite
number of homogeneous polynomials f
1
, f
2
,..., f
s
Πk[X
1
,X
2
,...,X
n+1
], so that
(
{
}
)
VS
()
=
V f f
,
,...,
f
s
.
12
Definition.
An ideal I Õ k[X
1
,X
2
,...,X
n+1
] is said to be
homogeneous
if for all f ΠI
all the homogeneous components of f belong to I.
(c)
Prove that an ideal I Õ k[X
1
,X
2
,...,X
n+1
] is homogeneous if and only if I is gen-
erated by a finite set of homogeneous polynomials.
Let
V
be a projective variety in
P
n
(k). Define
Definition.
()
=Œ
[
{
]
}
I
V
f
k XXXf
n
,
,...,
vanishes at every point of
V
.
12
+
1
(d)
Prove that I(
V
) is a homogeneous ideal.
It follows from (a)-(d) that the definitions in this exercise agree with the correspon-
ding definitions in this chapter.
10.8.6.
This exercise deals with some properties of the Zariski topology.
(a)
Show that if
U
1
and
U
2
are two nonempty open subsets of a variety
V
, then
U
1
«
U
2
π f. It follows that the Zarisk i topology is not Hausdorff. Also, every open
subset of a variety is dense in it.
(b)
Show that any infinite subset of a plane curve is dense in the curve.
Let
V
be a variety in k
n
. Using the notation defined by equations (10.23), show
that the projective completion of
V
in
P
n
(k) relative the coordinate system
(
U
n+1
,j
n+1
) is the closure (in the Zariski topology) of j
n+1
-1
(
V
) in
P
n
(k).
(c)
10.8.7.
(a)
Let f Πk[X
1
,X
2
,..., X
n
]. Show that if I =<f >, then H(I) =<H(f)>.
Consider the variety
V
= V(Y - X
2
,Z - X
3
) in k
3
. Show that
(b)
2
3
[
]
I
=
I
()
=<
V
Y
-
X
,
Z
-
X
>
and
ZW
-
XY
Œ
H I
()
Ã
k X Y Z W
,
,
,
,
but
(
)
(
)
>
2
3
ZW
-
XY
< -
H Y
X
,
H Z
-
X
.
This example from [Fult69] shows that (a) does not generalize, namely, if I =
<f
1
,f
2
,...,f
s
>, then it is not necessarily true that H(I) =<H(f
1
),H(f
2
),...,H(f
s
)>.
Section 10.9
10.9.1.
Find an implicit equation for the parameterized curve
2
xtt
yt
=+
=- +
2
t
