On duality on finite projective planes

Question

In nearly all (if not all) projective geometry texts I have bumped into the following theorem:

"Principle of duality: If in a theorem in $\mathfrak{P}$ one switches the word point for the word line and the corresponding incidence relations once again one obtains a theorem of $\mathfrak{P}$."

So far so good. Then I found this awesome list by G. Eric Moorhouse: http://www.uwyo.edu/moorhouse/pub/planes/ (Wayback Machine)

I noted the distinction between a Hall Plane and its dual. So looking a bit into the matter I kept running into the claim "Hall Planes are non-Desarguian and non-self-dual" and the modified version of the principle of duality, which claims the dualized theorem is true on the dual plane (this makes much more sense in my mind).

My question is twofold:

1. How does one prove that Hall Planes are not self dual? I haven't managed to find a proof of this fact!
2. What would be the true duality principle? If duality holds in the dual plane it should not hold in Hall Planes (as they're not self dual), yet all texts I've read claim that duality holds in any projective plane.

Thanks in advance!

P.S. Any good references on the concept of duality in projective geometry from a geometrical point of view would be much appreciated!

Answer 1

I'd expect that, in the duality principle that you quoted from "most (if not all) projective geometry texts", the symbol $\mathfrak P$ refers to the theory of projective planes, not to an arbitrary particular projective plane. One reason for this expectations is that theories, not planes, are the sort of entity that can "have" theorems (planes can satisfy statements, including theorems). Another reason is the observation you noted in your question; it's possible for a projective plane to satisfy a statement but not the dual statement. A final reason in favor of my expected interpretation of $\mathfrak P$ is that it makes the principle of duality true.

Answer 2

Your statement of the principle of duality is wrong, as Andreas has noted. What it says is that, if you have a theorem then the dual theorem, which you get by swapping points and lines, is also a theorem. It is very easy to derive incorrect results by applying the duality principle with too much enthusiasm.

The Hall planes are one of many classes of planes that are not isomorphic to their duals. Examples can be constructed as follows. Let $V$ be a vector space of dimension $d$ over $GF(q)$ and let $S$ be a set of $q^d$ matrices chased so the the difference of any two distinct elements of $S$ is invertible. Now construct an incidence structure with the elements of $V\oplus V$ as its points, and with the sets $$ L_{A,b} = \{(x,Ax+b): x\in V\} $$ as lines. Here $A\in S$ and $b\in V$. This structure is an affine plane, a so-called translation plane, and there is a very large literature devoted to them.

One example arises by choosing the matrices in $S$ so that they form the extension field of $GF(q)$, with order $q^d$. (In which case we get the classical Desarguesian plane.) But there are many other examples, and in general they are not self dual.

In fact there is a theorem that a translation plane is self dual if and only if it can be constructed from a set $S$ which is a semifield, that is, which satisfies all axioms for a field except associativity. Most translation planes do not come from semifields.

The most accessible treatment of this is still probably the book "Projective Planes" by Hughes and Piper. Note that my semifields are their division rings. Note also that I wrote "most accessible", not "accessible".

Answer 3

See here for a survey: http://en.wikipedia.org/wiki/Duality_(projective_geometry)

A true duality principle might be that a theorem for some projective plane induces the dual theorem for its dual. Mostly one is only concerned with projective geometries over fields, which are always self-dual.

One way to see that the Hall planes are not self-dual is that the defining quasifields are not semifields (they do not satisfy both distributivity laws), as being self-dual would imply the quasifield being a semifield.

Answer 4

What is duality, essentially? Well, when you consider a projective plane $P$ of the form $P=\mathbb{P}^2(\mathbb{K})$, where $\mathbb{K}$ is a field, then a point $p \in P$ is the base locus of a pencil of lines in $P$, namely a line in the dual space $P^*$. Conversely, given a line in $P$, it is a point on the dual space $P^*$ by definition. So you obtain a natural correspondence between points in $P$ and lines in $P^*$, and conversely.

Fixing a system of homogeneous coordinates $[x:y:z]$, you can see the construction above as a (non-canonical) projective isomorphism between $P$ and $P^*$, obtained by sending the point $[a: b: c] \in P$ to the line $\{ax+by+cz=0\} \in P^*$.

So, duality is naturally appearing when one considers projective planes obtained from $3$-dimensional vector spaces, but there is no reason that it should hold for completely general projective planes. And, in fact, it does not hold in general, as shown by the examples provided in the other answers.