Algebraic Geometry - Computer Graphics and Geometric Modeling - page 789

where m i is the multiplicity of the ith singularity and we sum over all singularities of
C . It turns out that the difference between the maximum number of double points of
a curve and actual number of double points defines an important invariant.

Definition. Let C be a plane curve in C 2 of order n defined by f(X,Y) = 0 that has
only ordinary singularities. The genus g of C is defined by

Â

( ) ( ) - ( ) gn n =- 12 - mm - ,

i i

i

where m i is the multiplicity of the ith singularity and we sum over all singularities.

For a general definition of the genus of a curve, even if it has nonregular singu-
larities, see, for example, [Walk50]. Note that inequality (10.90) implies that the genus
is a nonnegative integer.

10.15.4. Theorem. The genus of an irreducible curve is a birational invariant.

Proof. See [Walk50].

Theorem 10.15.4 has lots of consequences. For example, any nonsingular cubic
curve in P 2 is not rational because it has genus 1 and the genus of P 1

is 0. One can

also conclude the following (see [Walk50] for details):

(1) There are an infinite number of birationally nonequivalent curves. One way
to show this is to consider the curves C m defined by

2

() =

(10.91)

YFX - 0 ,

where F(X) is any polynomial of odd degree n = 2m + 1 with no multiple roots.
The curve C m is nonsingular. The nonsingular projective version of C m is called
a hyperelliptic curve . One can show that the hyperelliptic curve has genus m. See
[Shaf94].

(2) No nonsingular irreducible plane curves are equivalent if they have different
orders, the only exception being the case where one is a line and the other a conic.

In general, curves with the same genus need not be birationally equivalent. There
is one case where they are, however.

10.15.5. Theorem. (Noether's Theorem) A plane curve admits a rational parame-
terization (and is birationally equivalent to a line) if and only if it has genus 0.

Proof. See [Walk50], [Hoff89], [Abhy90], [Harr92], or [Shaf94].

It follows that any nonsingular cubic curve in P 2

is not rational because it has

genus 1.