Graphics Reference
In-Depth Information
(
)
=
fX X
,
,...,
X
X
,
12
n
+
1
i
is then referred to as the “plane at infinity” with respect to (
U
i
,j
i
). The coordinate neigh-
borhood (
U
n+1
,j
n+1
) represents the “standard” imbedding of k
n
in
P
n
(k). Although the
coordinate neighborhoods (
U
i
,j
i
) are quite adequate for most situations, they are not for
algebraic geometry. An arbitrary subvariety of
P
n
(k) might have lots of interesting points
lying on its plane at infinity. It would be much more convenient to be able to choose a
parameterization with respect to which all of those points, or at least all but a finite
number, are affine points. In addition, the choice of a coordinate system clearly has a big
influence on the equation for a set, so that the more choices one has, the better off one is.
For example, the standard coordinate axes of the projective plane are defined by
XY = 0,
but they could also be defined by
2
2
XY
-=
0
in another coordinate system. Therefore, we need to rethink our representation of
projective space.
From now on, let us think of
P
n
(k) as an abstract topological space with its defi-
nition inducing only one out of many possible homogeneous coordinates for its points.
In fact, let A = (a
ij
) be a nonsingular (n + 1) ¥ (n + 1) matrix over k and let
n
+
1
n
+
1
Tk
:
Æ
k
be the associated linear transformation that sends
x
to
x
A. The map T induces a well-
defined map
[]
n
n
()
Æ
()
T
:
P
k
P
k
defined by [T]([
c
]) = [T(
c
)]. Let T
i
be the ith component function of T defined by
()
=
(
()
()
()
)
TTT
c
c
,
c
,...,
n
c
.
1
2
+
1
Then
()
HH
=
=
ker
T
(10.20)
Ai
,
Ti
,
is a hyperplane in k
n+1
. Finally, let
n
()
-
UUP H
=
=
k
(10.21)
Ai
,
Ti
,
Ai
,
and define
n
jj
=
:
U
Æ
k
Ai
,
Ti
,
Ai
,
by
