Examples of curves

A curve is an ``absolutely irreducible projective curve defined over a field field''. Such a curve is determined by a polynomial $f(x,y)\in \mathbb{F}[x,y]$ of degree $d$ which is irreducible over $\overline{\mathbb{F}}$ . The projective curve $C$ is composed of three ``affine components''

$$C_1: f(x,y)=0,$$

$$C_2: g(u,v)=0,$$

$$C_3: h(w,z)=0,$$

where

$$g(u,v)=v^d f(1/v,u/v), \ h(w,z)=w^d f(z/w,1/w).$$

The projective version of $C$ is

$$z^d f(x/z,y/z)=0.$$

We shall sometimes abuse notation and confuse $C$ and $C_1$ .

The Klein quartic is given by $f(x,y)=x^3y+y^3+x$ . The projective version is $x^3y+zy^3+xz^3=0$ . (You can compute $f$ , $g$ , and $h$ above by setting $z=1$ , $y=1$ , and $x=1$ , respectively, in the projective version.)

The Fermat curve (of degree $n$ ) is given by $f(x,y)=x^n+y^n-1$ . The projective version is $x^n+y^n=z^n$ .

The Hermitian curve is a special case of the Fermat curve. It is defined over $\mathbb{F}=\mathbb{F}_{q^2}$ and is given by $f(x,y)=x^{q+1}+y^{q+1}+1$ .

Exercise 6.4.1 (a)   Compute $f$ , $g$ , and $h$ for the Klein quartic.

(b) Compute $f$ , $g$ , and $h$ for the Fermat curve.

(c) Compute $f$ , $g$ , and $h$ for the Hermitian curve.