Graphics Reference
In-Depth Information
CHAPTER 10
Algebraic Geometry
Prerequisites:
Chapter 3, 4, Sections 8.3-8.5, 8.11, 8.14, and basic abstract algebra
10.1
Introduction
Smooth manifold-like spaces are typically presented in one of two ways: either via an
explicit parameterization or as the set of zeros of some function. The definition of a
manifold was based on a parameterization, but special cases of manifolds, most
notably the conic sections, are described most easily by equations. The functions
involved are usually polynomials or at worst rational functions, which are quotients
of polynomials. Polynomials define ideals in polynomial rings and so algebraic struc-
tures become associated to implicitly defined spaces. The question then arises as to
whether there is any connection between purely algebraic properties of substructures
of polynomial rings (a subject that belongs to the field of commutative algebra) and
geometric properties of associated spaces. This is what algebraic geometry is all about.
Algebraic geometry can be thought of as the study of commutative algebra (the “alge-
braic” part of the name) as seen through the eyes of a geometer (the “geometry” part).
It is another attempt to study geometric properties using algebra, just like algebraic
topology.
The definitions below introduce the sets and the associated terminology that is
basic to algebraic geometry. These definitions and others along with various results
in this chapter are naturally expressed in the context of arbitrary fields k and subsets
of k
n
. However, it turns out that the best way to understand what is going on is to
analyze what happens in the case of algebraically closed fields first. What sets these
fields apart from others is that every polynomial of degree n over such a field has n
roots, so that sets defined as zeros of polynomials will always have the “right” number
of points. This explains why most theorems in algebraic geometry deal with alge-
braically closed fields. Drawing the desired conclusions back down at the original field
level is an additional and usually nontrivial step. Although we shall phrase much of
the discussion in this chapter in terms a general “field k” or “algebraically closed field
k,” to keep things concrete, the reader can always interpret this to mean either
R
or
