4 - Flocks of circle planes

Summary

Abstract

Flocks of finite circle planes—inversive, Minkowski and Laguerre planes—are surveyed, including their connections with projective planes, generalised quadrangles and ovals.

Circle planes

In the last thirty years, there has been considerable activity in the study of flocks of circle planes, originally by Thas, Walker and Fisher, but later, after Kantor, Payne and Thas had established connections between flocks of Laguerre planes and generalised quadrangles in the 1980s, by many authors. Their importance lies mainly in their connections with projective planes and generalised quadrangles.

The circle planes are the inversive, Minkowski and Laguerre planes, defined below. Their study received impetus when Benz published his book [9] devoted to them in 1973. They are related to ovoids, sharply 3-transitive sets and ovals, respectively.

Inversive planes

An inversive plane, I, is an incidence structure with a finite number of points and circles with the following properties.

1. (1) Every 3 distinct points are incident with a unique circle.

2. (2) Every circle has n + 1 > 2 points incident with it.

3. (3) There are n² + 1 points.

The integer n is called the order of I.

Example 1.1 The classical inversive plane I(q) has as its points the points of an elliptic quadric E of PG(3, q) and as its circles the non-tangent plane sections of E. It has order q, and automorphism group PΓO⁻(4, q). See [27] for more on elliptic quadrics. □