There are three partial quadrangles which arise from particularly nice caps of projective spaces, but there is one curious possibility left over. This missing piece in the puzzle was uncovered by Rob Calderbank almost thirty years ago. First I will explain how we might obtain a partial quadrangle from linear representation.
Suppose we have a geometry of points and lines which form a partial linear space of order ( points on any line and lines incident with any point). A so-called linear representation of into an affine space is an embedding such that the line set is mapped to a union of parallel classes of lines of . These lines of then define in the hyperplane at infinity a set of points of size (by intersection).
Example (Linear representations of ovoids of PG(3,q)):
Consider an ovoid of , and embed into as a hyperplane . This gives us an affine space , where the points of are the points of not in , and the lines of are the lines of not contained in . So there are affine points. Now consider the following geometry:
- Points: affine points
- Lines: affine lines meeting in a point of .
This produces a partial linear space of order . Moreover, it is a partial quadrangle known as and was first constructed by Peter Cameron.
More generally, if is a set of points of and if its linear representation is a partial quadrangle, then it is not difficult to prove that has to be a -cap of such that there is a constant such that each point not on is incident with tangents. Calderbank (1982) almost gave a complete classification of partial quadrangles which arise from linear representation. (Based on number-theoretic techniques!). Moreover, he listed the possible parameters of the associated strongly regular graph:
- n = 3 and is an ovoid of (as above);
- is an 11-cap of : s = 2, t = 10, . (There is an example due to Coxeter).
- is the unique 56-cap of : s = 2, t = 55, . (First discovered by Segre).
- is a 78-cap of : s=3, t=77, . (There is an example by Hill).
- is a 430-cap of : s=3, t=429, . (No such cap has yet been found).
Tsanakis and Wolfskill (1987) completed Calderbank’s classification by proving that if , then is an ovoid of .