Graphics Reference
In-Depth Information
where Y
i
= X
i
/X
n+1
are considered indeterminates.
Definition.
Let
V
be a projective variety in
P
n
(k). A
rational map
f:
V
Æ
P
m
(k) is a
map of the form
[
]
() ()
()
p
Æ
f
p
,
f
p
,...,
f
m
p
1
2
+
1
where the f
i
are homogeneous polynomials in k[X
1
,X
2
,...,X
n+1
] of the same degree
and at least one of the f
i
must be nonzero at every point
p
Œ
V
. The map f will be
denoted by the tuple (f
1
,f
2
,...,f
m+1
)
Clearly, two rational maps
=
(
)
=
(
)
m
()
fff
,
,...
f
,
ggg
,
,...,
g
:
VP
Æ
k
12
m
+
1
1 2
m
+
1
are equal if and only if f
i
g
j
= g
i
f
j
on
V
for all i and j. Since, given a rational map
f = (f
1
,f
2
,...,f
m+1
), we could divide through by one of the f
i
, we see that a rational
map is defined by m + 1 rational functions on
V
. Projections, defined below, are good
examples of regular rational maps.
Definition.
Assume that
X
is a d-dimensional linear subspace of
P
n
(k) defined the
(n - d) equations
LL
===
...
L
nd
=
0
,
1
2
-
where the L
i
are linear homogeneous polynomials. Define
n
()
Æ
n
--1
d
()
p
X
:
P
k
P
k
by
()
=
[
()
()
()
]
p
X
p
LL
p
,
p
,...,
p
.
1
2
nd
-
If
V
Õ
P
n
(k), then p
V
=p
X
|
V
is called the
projection of
V
with center
X
.
The map p
X
is clearly a regular map on
P
n
(k) -
X
. In fact, if
V
is any projective
variety that is disjoint from
X
, then p
X
|
V
is a regular rational map. To get a feel for
what the map p
X
does, let
Y
be any (n - d - 1)-dimensional linear subspace of
P
n
(k).
Then p
X
maps
p
Œ
P
n
(k) to the (unique) intersection of the linear subspace of
P
n
(k)
generated by
p
and
X
with
Y
. See Figure 10.20 for the case where d = 0.
Here is another example of a regular rational map.
Definition.
Fix n and d and define
n
()
Æ
N
()
v
:
P
k
P
k
d
by the condition
[
]
=
[
]
vxx
,
,...,
x
...,
m
,... ,
d
12
n
+
1
I
where m
I
ranges over all monomials of degree d in x
1
, x
2
,..., x
n+1
, of which there are
N = (
n +
d
) - 1. The map n
d
is called the
Veronese imbedding
of
P
n
(k) in
P
N
(k) and its
image n
d
(
P
n
(k)) is called the
Veronese variety
.
For example, if n = 2, then
