Graphics Reference
In-Depth Information
2
2
2
ZY
X
-
ZY
XZ Y
-
+
X
XZ Y
X
ZY
=
=
=
,
(
)
(
)
+
+
which is well defined at [0,1,1]. This shows that f is regular everywhere. The rest is
left as Exercise 10.13.8.
It is the fact that we were forced to figure out a way to rewrite the formula for the
function f in Example 10.13.31 over the open set
U
1
that shows that determining whether
or not a function is regular would not be any simpler if we were to define regular maps in
terms of tuples of homogeneous polynomials where common zeros are allowed.
Like in the affine case, a regular map
f:
VW
Æ
defines a homomorphism
[]
Æ
[]
f
*:
k
WV
k
.
Definition.
A regular map f :
V
Æ
W
between two projective varieties is called an
iso-
morphism
if it has an inverse that is also a regular map.
Next, we define rational functions between projective varieties. We cannot simply
define a rational function on a projective variety to be a regular function, because of
the following result:
10.13.32. Theorem.
A regular function defined on an irreducible projective variety
is constant.
Proof.
See [Shaf94].
Definition.
Let
V
be a projective variety in
P
n
(k). A
rational function
on
V
is a rational
homogeneous polynomial in k(X
1
,X
2
,...,X
n+1
). The set of rational functions on
V
is
called the
function field
of
V
and is denoted by k(
V
). A rational function h is
regular at
a point
p
Œ
V
, if it can be written in the form h = f/g, where f and g are homogeneous
polynomials of the same degree and g(
p
) π 0. In that case, h(
p
) = f(
p
)/g(
p
) is called the
value
of h at
p
. The set of regular points of h is called the
domain
of h.
The function field of a projective variety is clearly a field. In fact, it is easy to see
that
()
ª
(
)
k
k YY
n
,
,...,
,
12
where Y
1
, Y
2
,..., Y
n
are indeterminates, because
X
X
X
X
X
X
Ê
Ë
ˆ
¯
1
2
n
f
,
,...
,
1
(
)
(
)
fX X
,
,...
X
fY Y
,
,...
Y
,
1
12
n
n
+
+
1
12
12
n
n
+
1
n
+
1
n
+
1
=
=
,
(
)
(
)
gX X
,
,...
X
X
X
X
X
X
X
gY Y
,
,
....
Y
n
1
,
Ê
Ë
ˆ
¯
12
1
1
2
n
n
g
,
,...
,
1
n
+
1
n
+
1
+
1
