Graphics Reference
In-Depth Information
The projection of P
n
(k) with center X.
Figure 10.20.
P
n
(k)
X
d
Y
n-d-1
p
p
X
(p)
2
()
Æ
5
()
v
:
P
k
P
k
2
is defined by
]
=
[
]
[
2
2
2
vxxx
,
,
xxxxxxxxx
,
,
,
,
,
.
21 2 3
12 13 23
1
2
3
It is easy to see that the Veronese variety is in fact a projective variety in
P
N
(k). Some-
times any variety isomorphic to it is also called a Veronese variety.
Let
V
and
W
be projective varieties and assume that
W
Õ
P
m
(k). A
Definition.
rational map
f:
VW
Æ
is a rational map
m
:
VP
()
f
Æ
k
with the property that f(
V
) Õ
W
. The map f is called
birational
and we say that
V
and
W
are
birationally equivalent
if f has an inverse g :
W
Æ
V
that is a rational
map.
10.13.33. Theorem.
The image of a projective variety under a rational map is a
projective variety.
Proof.
See [Shaf94].
10.13.34. Theorem.
Any two nonsingular projective curves that are birationally
equivalent are isomorphic.
Proof.
See [Shaf94].
10.13.35. Corollary.
Any two nonsingular projective curves that are birationally
equivalent are homeomorphic.
We generalize the notion of a finite map to projective varieties by making it a local
property.
