Graphics Reference
In-Depth Information
Definition.
A regular map f :
V
Æ
W
between projective varieties is called a
finite
map
if every point
q
in
W
has an affine neighborhood
B
q
in
W
, so that
A
q
= f
-1
(
B
q
) is
an affine set in
V
and
f
AA B
:
Æ
q
q
q
is a finite map between affine varieties.
Projections are important examples of finite maps.
10.13.36. Theorem.
Let
V
be any projective variety in
P
n
(k) that is disjoint from a
d-dimensional linear subspace
X
in
P
n
(k). Then the projection of
V
nd
--
1
()
p
V
:
VP
Æ
k
with center
X
defines a finite map
()
V
Æ
p
V
.
V
Proof.
See [Shaf94].
A very useful application of Theorem 10.13.36 is the following generalization
needed later.
10.13.37. Theorem.
Let p
1
, p
2
,..., p
s+1
Πk[X
1
,X
2
,...,X
n+1
] be homogeneous poly-
nomials of degree d. If the p
i
have no common zeros on a projective variety
V
Õ
P
n
(k),
then
[
()
=
[
()
()
()
]
Œ
s
()
f
x
p
x
,
p
x
,...,
p
x
P
k
1
2
s
+
1
defines a finite map f :
V
Æ f(
V
).
Proof.
Consider the Veronese imbedding
n
()
Æ
N
()
v
:
P
k
P
k
.
d
Now, if
Â
i
i
i
(
)
=
pX X
,
,...,
X
a
X X
...
X
12
n
+
1
12
n
+
1
i i
...
i
12
n
+
1
12
n
+
1
i
++ =
...
i
d
1
n
+
1
is a homogeneous polynomial of degreed, then let
L
p
be the hyperplane of
P
N
(k)
defined by the linear equations
Â
a
X
=
0
,
ii
...
i
ii
...
i
12
n
+
1
12
n
+
1
i
++ =
...
i
d
1
n
+
1
where X
i
1
i
2
...i
n+1
is the indicated indexed variable in the collection, X
1
, X
2
,...,X
N
.
The property of the Veronese imbedding that we want to use here is that v
d
(V(p)) =
v
d
(
P
n
(k)) «
L
p
. To prove our theorem, let p be the projection of
P
N
(k) defined by the
hyperplanes
L
p
i
. One can show that f =p
°
v
d
, so that our theorem now follows from
Theorem 10.13.36.
