§ Decomposition of projective space
Projective space . The
current way I think about this is as follows (specialize to )
There's something awkward about this whole thing, notationally speaking. Is there
a more natural way to show that we have spent the projectivity to renormalize
- Consider a generic point . Either or .
- If , then we have which can be rescaled freely: . So, we get a component of from the .
- If , we have . Spend the projectivity to get . Now we have two free parameters, . This gives us the .
§ Projective plane in terms of incidence
We can define to be an object such that:
- Any two lines are incident at a single point.
- Two distinct points must be incident to a single line. (dual of (1))
§ The points at infinity
This will give us a copy of , along with "extra points" for parallel
- Consider two parallel lines and . These don't traditionally meet, so let's create a point at infinty for them, called .
- Now consider two more parallel lines, and . These don't traditionally meet either, so let's create a point at infinite for them, called .
- Finally, create another point as the point of intersection between and .
We can make a definition: the point at infinity for a given direction is the
equivalence class of all lines in that direction.
- Now, consider . We claim that they must all be equivalent. Assume not. Say that .
- Then there must a line that joins an . Call it . Now, what is the intersection between and the line ? The points and both lie on the line . But this is a contradiction: two lines must be incident at a single unique point.
- So we must have . So, for each direction, we must have a unique point where all lines in that direction meet.
§ The line at infinity
This now begs the question: what lines to different points at infinity lie on?
Let's consider as four points at infinity for four different
- Consider the lines that is incident on and , and then the line that is incident on the lines and .
- This begs the question: where do these lines meet? If we say that the meet at more new points of intersection, like this process will never end.
- So we demand that all points at infinity lie on a unique line at infinity.