Complex Numbers and Curves

The fundamental theorem of algebra—that a polynomial of degree

k

has exactly

k

complex roots—enables us to get the “right” number of intersections between a curve of degree

m

and a curve of degree

n

. However, it is not enough to introduce complex coordinates: getting the right count of intersections also requires us to adjust our viewpoint in two other ways.

1.

We must count intersections according to their

multiplicity

, which amounts to counting a root

x

=

r

of a polynomial equation

p

(

x

) = 0 as many times as the factor (

x

-

r

) occurs in

p

(

x

).

2.

We must view curves

projectively

, so that intersections at infinity are included.

For these reasons, and others, algebraic geometry moved to the setting of complex projective space in the 19th century. In this chapter we see how this viewpoint affects our picture of algebraic curves. The simplest such curve is the

complex projective line

, which turns out to look like a sphere. Other algebraic curves also look like surfaces, but they can be more complicated than the sphere. It was discovered by Riemann that

rational curves

(curves that can be parameterized by rational functions) are essentially the same as the sphere, but nonrational curves have “holes” and hence are essentially different. This discovery reveals the role of

topology

in the study of algebraic curves.