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when expressed in homogeneous coordinates relative to A.
Proof.
The proof is easy. (The reader may also find it helpful to review the discus-
sion in Section 6.10 in [AgoM05] about the way that equations for implicitly defined
objects transform.)
10.3.4. Example.
To find a coordinate neighborhood (
U
,j) for
P
2
, so that the line
defined by the equation -2X + Y - Z = 0 becomes its line at infinity.
Solution.
Consider the linear transformation
TX Z
Y Y Z
ZXYZ
:
¢=
-
¢=
-
¢=-
2
+
-
It is nonsingular because the inverse of T is easily found by back substitution to be
1
2
1
2
-
1
TX
:
¢=
YZ
-
YXY
ZX
¢=-
+
¢=-
Note that T maps the points (X,Y,Z) with -2X + Y - Z = 0 onto the set of points
(X¢,Y¢,Z¢) with Z¢=0. Therefore, we can let
U
=
U
T, 3
and j=j
T, 3
:
U
Æ
R
2
, that is,
Z
XYZ
-
-+-
YZ
XYZ
Ê
Ë
ˆ
¯
(
[
]
)
=-
-+-
j XYZ
,,
,
.
2
2
The important consequence of Theorem 10.3.3 is that if two coordinate systems
for projective space differ by a linear change of variables, then the polynomials that
define a variety with respect to these coordinate systems only differ by a linear change
of variables. Since a linear change of variables will never affect any of the properties
of varieties we are interested in, it follows that by choosing an appropriate homoge-
neous coordinate system we can reduce problems about properties of points on hyper-
surfaces in
P
n
(k) to problems about points on hypersurfaces in k
n
. More generally, we
shall feel free to choose appropriate coordinate systems for k
n
because there is a
natural correspondence between linear changes of coordinate systems there and in
P
n
(k). These comments justify future phrases like “Without loss of generality assume
a coordinate system so that. . . .”
We have indicated that projective space is the natural space in which to do alge-
braic geometry. However, many problems arise from an attempt to understand affine
varieties. This leads to another problem. How do we convert an affine variety into a
projective one? Before we address this question we need some more definitions.
Consider a polynomial f(X
1
,X
2
,...,X
n
) in k[X
1
,X
2
,...,X
n
]. We express f in terms
of its homogeneous components. If f has degree d, then f can be written uniquely in
the form