Examples of points and divisors

The set of all rational points of $C$ is the set

$$C(\mathbb{F})=C_1(\mathbb{F})\cup C_2(\mathbb{F})\cup C_3(\mathbb{F}),$$

where

$$C_1(E)=\{(x,y)\in E^2 \vert f(x,y)=0\},$$

$$C_2(E)=\{(u,v)\in E^2 \vert g(u,v)=0\},$$

$$C_3(E)= \{(w,z)\in E^2 \vert h(w,z)=0\},$$

and $E$ is any extension of $\mathbb{F}$ . The coordinate map $u=y/x, v=1/x$ maps $C_1$ with $x\not= 0$ to $C_2$ , and the coordinate map $w=1/y, v=x/y$ maps $C_1$ with $y\not= 0$ to $C_3$ . These give rise to coordinate maps $\phi:C_1\rightarrow C_2$ , $\phi':C_1\rightarrow C_3$ , Generally, there are coordinate maps $\phi_{ij}:C_i\rightarrow C_j$ , for each $i\not= j$ , obtained by composing these maps $\phi$ and $\phi'$ and their inverses.

Of course, if a point $P$ of one component is sent to a point $P'$ in a different component by one of these coordinate maps then we regard $P$ and $P'$ to be the same element of $C(\mathbb{F})$ .

The concept of ``smoothness'' is a restriction on the function $f$ : we call $C$ smooth (over $\overline{\mathbb{F}}$ ) if

$$(\frac{\partial f}{\partial x} (x,y), \frac{\partial f}{\partial y} (x,y))\not= (0,0),$$

for all $(x,y)\in C_1(\overline{\mathbb{F}})$ , and

$$(\frac{\partial g}{\partial u} (u,v), \frac{\partial g}{\partial v} (u,v))\not= (0,0),$$

for all $(u,v)\in C_2(\overline{\mathbb{F}})$ ,

$$(\frac{\partial h}{\partial w} (w,z), \frac{\partial h}{\partial z} (w,z))\not= (0,0),$$

for all $(w,z)\in C_3(\overline{\mathbb{F}})$ .

Example: The point $p=(3,2)$ is a rational point of the Klein quatric over $\mathbb{F}_5$ .