Graphics Reference
In-Depth Information
ff f
=++ +
01
...
d
,
where f
i
is a homogeneous polynomial of degree i in k[X
1
,X
2
,...,X
n
].
Definition.
The polynomial
(
)
=
d
d
-
1
FX X
,
,...,
X
fX
+
fX
+
...
+
f
X
+
f
12
n
+
1 0
1
dn d
-
1
+
1
n
+
1
n
+
1
in k[X
1
,X
2
,...,X
n+1
] is homogeneous of degree d. It is called the
homogenization of f
and is denoted by H(f).
10.3.5. Example.
If
3
2
(
)
=
f X Y
,
25
X
-
XY
+
XY
-
7
,
then the homogenization of f is
(
)
=
3
2
3
F X Y Z
,,
25
X
-
XYZ
+
XY
-
7
Z
.
Another way of looking at this process is that F(X,Y,Z) is obtained by replacing X and
Y in f(X,Y) by X/Z and Y/Z, respectively, and then clearing denominators.
Definition.
If F(X
1
,X
2
,...,X
n+1
) is a homogeneous polynomial in k[X
1
,X
2
,...,X
n+1
],
then the polynomial
)
Œ
[
]
(
)
=
(
f XXXF XXXk XXX
,
,...,
,
,...,
,
1
,
,...,
12
n
12
n
12
n
is called the
dehomogenization
of F and is denoted by D(F).
Taking another look at Example 10.3.5 we see that f(X,Y) = F(X,Y,1). In general,
D(H(f)) = f but H(D(F)) does
not
always equal F as is shown by the example F = X
1
X
2
and n = 2.
Definition.
Let
V
be an (affine) variety in k
n
. Using the notation defined by equa-
tions (10.23), the smallest projective variety in
P
n
(k) which contains j
-
n+1
(
V
) is called
the
projective completion
of
V
relative the coordinate system (
U
n+1
,j
n+1
) and is denoted
by H(
V
). More precisely,
H(
V
) = the intersection of all projective varieties in
P
n
(k) that contain j
n+1
-1
(
V
).
If
V
is a (projective) variety in
P
n
(k), then
V
«
U
n+1
is called the
affine part
of
V
and
will be denoted by D(
V
).
It is the operations of homogenization, dehomogenization, projective completion,
and affine part that allow us to pass back and forth between varieties in affine and
projective space. See [Kend77] for more details on the relationship between affine and
projective varieties. In particular.