What is the point of having at least three points on every line of a projective plane?

Question

A common definition of "projective plane" includes the following axioms:

1. every pair of distinct points lies on exactly one line
2. every pair of distinct lines meet in exactly one point
3. every line has at least three points
4. not all points are on the same line

(taken from the nlab)

I wonder: What's the point of having the axiom 3.?

I suppose the answer is "to exclude 'degenerate' cases", however I want to see more "practical" reasons. Do many important theorems about projective planes fail for these cases?

If somebody knows about a category or categories of projective planes (morphisms could be "taking lines to lines" for example) and about, how including or excluding the axiom 3. affects the properties of such a category, it would be interesting to me aswell.

Answer 1

I will try to provide an answer not along the pathologies that arise when removing axiom 3, but rather what a useful generalization should be.

Let me first remind you of the Veblen-Young Theorem:

If Desargues theorem holds in an abstract projective plane, it is of the form $P(V)$ for some vectorspace $V$ over a skew-field $k$. If Pappus theorem holds as well, the field can be found to be commutative.

This is a form of concreteness result. In fact if we allow ourselves to talk about abstract projective spaces of general dimension, all abstract projective spaces of dimension greater than $2$ are of the form $P(V)$ for a vectorspace $V$ over a skewfield. This obviously requires axiom 3, since it holds for all projective spaces of the form $P(V)$!

As an example, the above theorem would obviously not hold if we allow as our projective plane a three element set with lines defined as two-element subsets. Both Pappus and Desargue are true in this 'projective plane'.

There is however a way of making sense of the above example: It "should" be a projective space over the "field with one element". Now note that such a thing in the strict sense of the wording doesn't exist. But in a lot of different areas in mathematics there's evidence that a generalization of the term field should exist and that there should be something people call 'absolute geometry', i.e. algebraic geometry over the field with one element.

If you want to see a definition of projective space that doesn't enforce axiom 3, but is still sensible, I recommend Cohn's introductory paper Projective Geometry over $\mathbb F_1$. Definition 1 in this paper may give you a more satisfactory feeling.

Answer 2

Without axiom 3. there is no longer true that any two lines have the same cardinal, that the number of points equals the number of lines, that the number of lines containing a point is constant and others in the same vein. An important example are affine planes (but also without axiom 2).

Regarding the last part part of the question, you need to look at the notion of incidence structure or incidence geometry. A decent place to start is the relevant wikipedia page.

Also J. Ueberberg, Foundations of Incidence Geometry is a nice introduction.

Answer 3

The real reason is to avoid degenerate cases. Sometimes the properties are given as:

1. Each two distinct points of P are incident with a unique line.
2. Each line is incident with at least three points.
3. Each two distinct lines of P meet in a unique point.
4. There are four points of P with no three of them collinear.

In other words, we want our projective plane to contain at least one quadrangle (note that my 2. and 4. together are equivalent to your 3. and 4. together). Eliminating this requirement, we can end up with either a triangle (3 points, 3 lines) or examples like the one given by adhalanay, where you start with a triangle and add extra points on one of the lines (and extra lines to complete everything). What are the big problems with these examples?

1. You no longer get an affine plane through the normal method (defining a line to be $\ell_{\infty}$ and removing it along with its points). Depending on how this is done, you would either end up with only a single point left, or you would end up with only a single line.
2. If you do not have the same number of points on every line, then the point/line adjacency graph no longer gives an example of a Moore graph, unlike in the other cases.
3. Problems come up in coordinatizing the plane with a Planar Ternary Ring.