Graphics Reference
In-Depth Information
(1) One can choose homogeneous coordinates for
P
n
(k) so that for some neigh-
borhood
U
of
p
in
V
the image f(
U
) belongs to the affine part of
P
n
(k) with
respect to these coordinates. With this choice we can (and will) consider the
map g = f|
U
as a map from
U
to k
n
.
(2) If g(
q
) = (g
1
(
q
),g
2
(
q
),..., g
n
(
q
)), then the g
i
:
U
Æ k are rational functions that
are regular at
p
.
In this case, we say that the function f is
regular at
p
. The function f is said to be
regular on
V
if f is regular at all points of
V
.
Again one can show that the notion of being regular, or regular at a point, does
not depend on n,
U
, or the homogeneous coordinates that are chosen. As we warned
earlier, defining regularity of maps between projective varieties is complicated. It
involves checking that for each point one can find a coordinate system with respect
to which the function is affine-valued and can be expressed in terms of regular rational
functions. One type of function that it is easily seen to be a regular map is one that
can be expressed globally as an (n + 1)-tuple of homogeneous polynomials of the same
degree that do not have any common zeros. It is unfortunate that, as the next example
shows, not all regular maps can be obtained in this way, because that would have
allowed for a much simpler definition.
Consider the variety
V
in
P
2
(
C
) defined by
10.13.31. Example.
2
2
2
XYZ
+-=
0
and the “stereographic projection”
1
()
f
fXYZ
:
,,
VPC
Æ
(
[
]
)
=
[
]
XZ Y
,
-
.
The polynomials X and Z - Y have a common zero at (0,1,1), so that f is not defined
at [0,1,1], but we define it there by setting f([0,1,1]) equal to [1,0]. We claim that f is
a regular map. Furthermore, it cannot be expressed as a pair of homogeneous poly-
nomials without common zeros.
Consider the open cover {
O
1
,
O
2
} of
P
1
(
C
), where
Proof.
=
[
{
]
}
=
[
{
]
}
O
xy x
,
π
0
and
O
xy y
,
π
0
.
1
2
Let
-
1
()
=
2
()
-
{
[
]
}
-
1
()
=
2
()
-
[
{
]
}
U
=
f
OP
C
01 1
,,
-
and
U
=
f
OP
C
011
,, .
1
1
2
2
Then f is regular on
U
2
, because f sends [X,Y,Z] to the point whose affine coordinate
is X/(Z - Y). On
U
1
the point [X,Y,Z] gets sent to the point whose affine coordinate is
(Z - Y)/X. It may seem as if there is a problem with regularity at [0,1,1], but for-
tunately we can rewrite the quotient as
