The projective plane

A cool way to think about a projective plane is that it is something that has points and lines which come endowed with an incidence correspondence (i.e., we know what it means for a point to be on a line) subject to the following axioms:

1. Given any two distinct points there exists a unique line passing through both of them.
2. Given any two distinct lines there exists a unique point which lies on both of them.

It turns out that given a field K we can find a particular projective plane, namely P^2(K) whose points correspond to 1-dimensional sub vector spaces of K^3. We will usually think of a point in P^2(K) as a triple (a : b : c) of elements of K, not all zero. Thus (a : b : c) defines the same point as (2a : 2b : 2c) provided that 2 isn't zero in K (which actually sometimes happens, right?). Anyway, a line of P^2(K) is typically given by an equation

• L : Ax_0 + Bx_1 + Cx_2 = 0

where A, B, C are elements of K, not all zero. We say that (a : b : c) lies on L if and only if

• Aa + Bb + Cc = 0

This of course means that L depends on (A, B, C) only up to a scalar. Hence we obtain the pleasing feature of this particular projective plane that the set of lines and set of points have the same cardinality.

Exercise 1: Prove that an axiomatic projective plane has the same number of points as lines

Exercise 2: Prove that the example P^2(K) as defined above is indeed a projective plane

Exercise 3: If K is a finite field of order q how many points does P^2(K) have?

Exercise 4: If K = Z/pZ write a script finding the intersection point of two lines

Exercise 5: If K = Z/pZ write a script finding the line passing through two given points

Exercise 6: Find other ways to think about lines in P^2(K) by parametrizing lines. 