Graphics Reference
In-Depth Information
The genus that we just defined is an algebraic concept, but if we are talking about
curves over the complex numbers, then Theorem 10.2.5 implies that the curve is either
a surface or a singular surface where a finite number of points have been identified.
In this case, the algebraic genus and the topological one which was defined in Chapter
6 are the same. (In the singular surface case consider the topological genus to be
defined by equations (6.3).) The proof of Theorem 10.15.5 does not really provide a
good algorithm for finding the rational parameterizations. There are simple algo-
rithms in the case of curves of order 2 or 3 or so-called
monoids
(a curve of degree n
with a point of multiplicity n - 1). See the discussion earlier in the section or the
lengthy discussion in [Hoff89].
The genus g is not the only birational invariant of a curve. There are continuous
invariants called “moduli.” Only when two curves have genus 0 are they isomorphic.
There are curves with the same genus g > 0 that are not isomorphic. See [Shaf94].
We finish this section with one final observation. If an implicitly defined set con-
sists of a finite number of points (so that a parameterization amounts to simply listing
the zeros), then there is an algorithm that will find those zeros. The algorithm involves
using Gröbner bases. See [Mish93].
10.16
The Dimension of a Variety
Every topological space has a notion of dimension associated to it. We defined
this concept in the case of cell complexes and manifolds. Since an algebraic variety
lives in projective space, it can be thought of as a topological space and so has a dimen-
sion. The interesting question is whether one can determine its dimension from its
algebraic structure and if yes, then how one would compute it.
Here are some simple two-dimensional examples of varieties V(f) in
R
3
.
10.16.1. Example.
f(X,Y,Z) = aX + bY + cZ + d
Description.
V(f) is an arbitrary plane that intersects the x-, y-, and z-axis at -d/a,
-d/b, and -d/c, respectively (if a, b, and c are nonzero).
f(X,Y,Z) = X
2
+ Y
2
+ Z
2
10.16.2. Example.
- 1
Description.
V(f) is the sphere of radius 1 about the origin.
2
2
2
X
a
Y
b
Z
c
(
)
=+--
10.16.3. Example.
fXYZ
,,
1
2
2
2
Description.
V(f) is a connected hyperboloid with horizontal slices that are ellipses.
See Figure 10.21(a).
2
2
2
X
a
Y
b
Z
c
(
)
=---
10.16.4. Example
.
fXYZ
,,
1
2
2
2
Description.
V(f) is a disconnected hyperboloid with vertical slices that are ellipses.
See Figure 10.21(b).
