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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
) Π[
]
(
) =
(
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.
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