## Examples of points and divisors

The set of all rational points of is the set

where

and is any extension of . The coordinate map maps with to , and the coordinate map maps with to . These give rise to coordinate maps , , Generally, there are coordinate maps , for each , obtained by composing these maps and and their inverses.

Of course, if a point of one component is sent to a point in a different component by one of these coordinate maps then we regard and to be the same element of .

The concept of smoothness'' is a restriction on the function : we call smooth (over ) if

for all , and

for all ,

for all .

Example: The point is a rational point of the Klein quatric over .

Subsections

david joyner 2008-04-20