Curves
We start with a naive definition.
Definition
A curve over a field k is the set of zeros in
k × k of a single polynomial equation in
two variables, f(x, y) = 0.
Examples
Let k=R and consider the following curves.
(Here we should insert the graphs of the following curves.)

y = 0.

y^{2} = 0.

xy = 1.

x^{2} + y^{2} = 1.

x^{2} + y^{2} = 0.

x^{2} + y^{2} = 1.

y^{2} = x^{3}.

y^{2} = x^{3} + x^{2}.

y^{2} = x^{3}  x.

x^{2}  y^{2} = 0.

(y  x)(y  x + 1) = 0.

x^{3} + y^{3} = 1.
These examples raise several important
questions about the
naive definition. For example
 Are (i) and (ii) the same or different curves?
 Should (v) and (vi) even be called curves at all?
 Why are (iii), (ix) and (xi) disconnected?
 What's special about (vii), (viii), and (x)?
The questions are of two kinds. The first two questions are
metaphysical; they ask if we have the correct definition. The final
two questions are analytic; they begin to study the intrinsic geometry
of curves.
To elaborate on question (A), consider the families of curves
(Pictures again need to be inserted.)
 y = ax.
 y^{2} = ax.
Intersect each family of curves with the line x =
1. When a ≠ 0, curves in the first
family intersect the line in exactly one point, but curves in the
second family intersect in two points. As the parameter a
→ 0, the limiting curve of the first family is
curve (i), and it still intersects in one point. However, the limiting
curve in the second family is curve (ii), and the two points of
intersection coalesce in the limit. Should the limit curve intersect
the line in one point, or in two points "properly counted"?
Regarding question (B), there are two schools of thought (which will
be described shortly). In either case, the following theorem shows
that the underlying difficulty is that R is not
algebraically closed.
Theorem Let k be an algebraically closed field and
let f ∈ k[x,y] be a nonconstant
polynomial. Then there are infinitely many solutions to
f(x,y)=0.
Proof: Write
f(x,y) = a_{0}(y) + a_{1}(y)x + … +
a_{n}(y)x^{n} , with
a_{n}(y) ≠ 0, as a
polynomial of degree
n in
x with
coefficients from
k[y]. If
n = 0,
then we simply have
f(x,y) = a_{0}(y), a nonconstant
polynomial in one variable. Since
k is
algebraically closed, this polynomial has a root
y_{0}. Then all of the infinitely many pairs
(x, y_{0}) as
x ranges over the
(infinite) algebraically closed field
k provide
solutions to
f(x,y) = 0.
So, suppose n > 0. Then a_{n}(y) =
0 has only finitely many solutions. There are infinitely
many points y_{0} where a_{n}(y_{0}) ≠
0; at each one of them, we get a polynomial
f(x,y_{0}) of degree n in the
single variable x. This polynomial has a solution
x_{0}, yielding infinitely many points
(x_{0},y_{0}) where f(x,y) = 0.
The restrictive school of thought says that you can avoid the
difficulties posed by examples (v) and (vi) by considering only
algebraically closed fields. This seems to be rather narrowminded,
since it foregoes any chance of applying geometric techniques to some
interesting problems that arise in number theory. A more openminded
approach is to "fix" the naive definition to make it useful over
arbitrary fields. Nevertheless, we will acquiesce (at least
temporarily), and work over algebraically closed fields for the
remainder of this chapter.
Definition
As a set, affine
nspace over a field
k is defined to be the Cartesian product
A^{n}_{k} = k × k × … × k
of
k with itself
n times.
Definition
Let k be algebraically closed. An
affine algebraic set over k is
the set of common zeros in
A^{n}_{k} of some set of polynomials S
⊂ k[x_{1}, …, x_{n}]. Given such a set
S, its algebraic set of zeros will be denoted
Z(S).
Definition
The Zariski topology on A^{n} is
defined by taking the closed sets to be the algebraic sets. The
Zariski toplogy on an algebraic set is defined as the topology induced
from its embedding in A^{n}.
Comments on this web site should be addressed to the
author:
Kevin R. Coombes
Department of Biomedical Informatics
The Ohio State University
Columbus, Ohio 43210