Graphics Reference
In-Depth Information
()
()
()
()
()
()
()
()
T
T
c
c
T
T
c
c
T
T
c
c
T
T
c
Ê
Ë
ˆ
¯
1
i
-
1
i
+
1
n
+
1
[
()
=
(10.22)
j
Ai
c
,...,
,
,...,
.
,
c
i
i
i
i
10.3.1. Proposition.
The maps j
A,i
are homeomorphisms and the (
U
A,i
,j
A,i
) are
coordinate neighborhoods that cover
P
n
(k).
Proof.
Exercise 10.3.1.
Definition.
The tuple T(
c
) will be called
homogeneous coordinates
of [
c
] in
P
n
(k)
relative to A
. The collection of coordinate neighborhoods
{
(
)
}
U
Ai
j
,,
Ai
,
is called the
coordinate neighborhood cover induced by A (or T)
. A point of
H
A,i
is called
an
ideal point
or
point at infinity
of
P
n
(k)
with respect to the coordinate neighborhood
(
U
A,i
,j
A,i
). The set
H
A,i
is called the
plane
(or
line
, if n = 2)
at infinity with respect to
the coordinate neighborhood
(
U
A,i
,j
A,i
).
If A is the identity matrix I, then we get our standard homogeneous coordinates,
neighborhoods, ideal points, and planes at infinity for
P
n
(k). It will be convenient to
abbreviate the above notation as follows:
HH
UU
=
=
=
,
,
.
i
I i
,
and
i
I i
,
(10.23)
jj
i
I i
,
In fact, a consequence of the next two theorems is that we may always assume,
without loss of generality, that the homogeneous coordinates we use correspond to
the most standard of all cases, namely, i = n + 1.
10.3.2. Theorem.
Given any hyperplane in
P
n
(k) we can always find a nonsingular
(n + 1) ¥ (n + 1) matrix A so that this hyperplane is the plane at infinity with respect
to the coordinate neighborhood (
U
A,n+1
,j
A,n+1
).
Proof.
The theorem is an immediate consequence of the fact that given any hyper-
plane
X
in k
n+1
we can find a nonsingular linear transformation of k
n+1
that maps
X
onto the hyperplane x
n+1
= 0.
When we defined projective hypersurfaces and varieties earlier, those definitions
implicitly assumed the
standard
parameterization of projective space. Although a
variety is a unique subset of projective space, how it is presented depends on the para-
meterization. The next theorem tells us how the polynomials that define a variety
change as we switch from one set of homogeneous coordinates to another.
10.3.3. Theorem.
Let
V
= V(f) be any hypersurface in
P
n
(k) defined by a homogeneous
polynomial f. If A is any nonsingular (n + 1) ¥ (n + 1) matrix, then
V
= V(g), where
(
)
(
)
=
(
)
-
1
g YYYf
,
,...,
YYYA
,
,...,
,
12
n
+
1
12
n
+
1
