Graphics Reference
In-Depth Information
The definition of a space curve will be based on parameterizations and is gotten by
expressing K as an extension of k by n elements, n > 2 . The extra elements in the
basis will let us interpret the curve as a curve in k
n
. This new way of looking at the
plane curve will tell us what the definition of a curve in k
n
should be.
After these preliminary comments, assume that
(
)
Kk
=
xx
,
,...,
x
12
n
is an arbitrary extension of k with transcendence degree 1. By Theorems B.11.4
and B.11.8, K will have the form k(x,y) described in the previous paragraph and
correspond to an affine plane curve
C
f
defined by an equation
(
)
= 0
fXY
,
.
Assume that f(X,Y) π aX + b , so that y π 0, which is the interesting case. Assume further
that the projective curve that it represents is irreducible. It follows that we can express
the x
i
in terms of x and y and conversely that we can express x and y in terms of the x
i
.
In other words, there exist g, h Πk(X
1
,X
2
,...,X
n
) and z
i
Πk(X,Y) so that
(
)
x
i
=
i
zxy
,
and
(
)
(
)
xg
=
xx
,
,...,
x
,
yh
=
xx
,
,...,
x
.
12
n
12
n
Any parameterization (x(t),y(t)) representing a place P
of
C
f
induces a
“parameterization”
(
(
() ()
)
(
( )
( )
)
(
() ()
)
)
=
(
()
()
()
)
zxtyt z xtyt
,
,
,
,...,
z xtyt
,
xtx t
,
,...,
x t
.
1
2
n
1
2
n
Switching to projective coordinates we get a “parameterization”
(
()
()
()
)
z
t z
,
t
,...,
z
t
,
where
x
i
=
z
z
.
1
2
n
+
1
i
n
+
1
Choose our representative for the projective point so that min(ord(z
i
(t))) = 0. Let
(
()
()
()
)
q
=
zz
00
,
,...,
n
0
.
1
2
+
1
Let
C
P
be the subset of
P
n
(k) consisting of all these points
q
as we let P range over
all places of
C
f
. Let
C
be the affine subset of k
n
associated to
C
P
.
Definition.
The subset
C
P
in
P
n
(k) that is obtained in this way is called an irreducible
projective space curve
. The subset
C
in k
n
is called an irreducible
affine space curve
.
Equivalence classes of parameterizations of
C
P
or
C
are called
places
for the
curves. All the terminology that we developed for places for plane curves apply here.
One can also define birational correspondences
(
)
Æ= ¢
(
)
uK k
:
=
qq
,
,...,
q
K k
qq
,
¢
,...,
q
¢
,
12
n
12
n
