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(1) (The Implicitization Problem) Given a polynomial parameterization p(t) of a
space
X
, can one find an implicit equation which defines
X
?
(2) (The Parameterization Problem) If a space
X
is defined implicitly by means of
a polynomial equation f(
p
) = 0, can one parameterize
X
using polynomials?
(3) (The Intersection Problem) Given implicit or parametric definitions for spaces
X
and
Y
, what can be said about their intersection
X
«
Y
?
Here is a look at what is to come in this chapter. After Section 10.2 gets us started
with some examples of plane curves that show why affine space is inadequate for a
thorough analysis of varieties, Sections 10.3-10.5 back up to fill in some needed alge-
braic background. Section 10.3 describes some analytic properties of the parameter-
ization of projective space and useful facts about how to pass back and forth between
affine and projective space. Section 10.4 defines the resultant of two polynomials and
shows how it can be used to find common factors. Section 10.5 describes some basic
algebraic properties of polynomials and their influence on the structure of varieties.
Sections 10.6 and 10.7 define intersection multiplicities, singularities, and tangents of
plane curves and use this to analyze their intersections. Next, as preparation for study-
ing higher-dimensional varieties, we develop some simple aspects of commutative
algebra in Section 10.8. Section 10.9 looks at the problem of finding implicit repre-
sentations of parametrically presented curves. Difficulties with using the resultant lead
to a discussion of Gröbner bases in Section 10.10 and elimination theory in Section
10.11. Section 10.12 starts the analysis of the singularities of a curve and defines the
place of a curve. This is a long section because in order to understand what is going
on we have to bring complex analysis into the picture, in particular, the topics of
analytic continuation, the uniformization problem, and Riemann surfaces. Section
10.13 is on rational and birational maps. We move on to higher dimensions and space
curves in Section 10.14. The parameterization problem for implicitly defined curves
is discussed in Section 10.15. We finish the chapter with an overview of some higher-
dimensional topics in Sections 10.16-10.18. We define the dimension of a variety,
describe the Grassmann manifolds as varieties, and sketch some important theorems.
A final note.
At various times in this chapter we will be taking derivatives or partial
derivatives of polynomials. Let us clarify this right now, so that the reader will not be
puzzled by what that might mean in the case of, say, polynomials over the complex
numbers or other fields. In the case of polynomials there is a formal notion of deriv-
ative that is computed like the usual derivative but is well defined for polynomials
over any ring and does not involve having to take limits. See Section B.7. This is what
we shall be using.
10.2
Plane Curves: There Is More than Meets the Eye
We begin our tour of algebraic geometry with a closer look at plane curves. Even
though the reader's initial reaction might be that their low dimensionality would not
lead to anything interesting, this is not at all the case. In fact, in contrast to algebraic
topology where things tend to get interesting only in higher dimensions, an analysis
of plane curves will already lead to some of the most fundamental ideas in algebraic
