Graphics Reference
In-Depth Information
10.14.5. Example.
Consider the space curve called the
twisted cubic
(
)
Ã
2
3
3
V
=- -
Vy x z x
,
R
.
Here are some facts about it:
(1) It can be parameterized by the map t Æ (t,t
2
,t
3
), so that it is the same curve
as the one defined in Exercise 9.4.2
(2) I (
V
) =<y - x
2
,z - x
3
> (See [CoLO97])
(3)
V
is irreducible (Theorem 10.8.14).
(4) The curve is an example that shows that homogenizing its equation does not
lead to the smallest projective variety containing it. See Exercise 10.3.4. On
the other hand, a Gröbner basis for it is
{
}
2
2
G
=- - -
x
y xy
,
z xz
,
y
,
and a homogenization of this basis does lead to a basis
{
}
2
2
xwy ywz zy
-
,
-
,
-
for the projective ideal for that smallest projective variety. See [CoLO97].
(5) Property (3), Theorem 10.8.12, and Theorem 10.14.4 imply that the twisted
cubic is a space curve.
The argument that showed that the twisted cubic is irreducible can be extended
to prove the following:
10.14.6. Theorem.
If k is an infinite field, then any affine variety
V
in k
n
that has a
rational parameterization is irreducible.
Proof.
See [CoLO97]. Theorem 10.8.14 was a special case of this theorem.
Applying Theorem 10.14.6 in the special case of polynomial parameterizations
gives lots of examples of irreducible varieties, so that, using Theorem 10.8.12 and
Theorem 10.14.4, we get lots of examples of space curves.
10.14.7. Theorem.
Every space curve is birationally equivalent to a plane curve that
is either nonsingular or has only ordinary singularities.
Proof.
See [Walk50] or [Seid68].
The fact that every space curve is birationally equivalent to a plane curve was
already suggested from Theorems 10.13.14 and 10.13.15. The only problem is that
they applied to hypersurfaces, which (nonplane) curves are not. Now, Theorem
10.13.27 showed that every space curve is birationally equivalent to a nonsingular
curve; however, this curve may
not
be planar. There are nonsingular space curves
that are not birationally equivalent to a nonsingular plane curve. We can reduce any
singularities to ordinary double points, however.
