Finite Laguerre Near-planes of Odd Order Admitting Desarguesian Derivations

From this definition it readily follows that a Laguerre plane of order n has n + 1 generators, that every circle contains exactly n + 1 points and that there are n3 circles. All known models of finite Laguerre planes are of the following form. Let O be an oval in the Desarguesian projective plane P2 = PG(2, pm), p a prime. Embed P2 into threedimensional projective space P3 = PG(3, pm) and let v be a point of P3 not belonging to P2. Then P consists of all points of the cone with base O and vertex v except the point v. Circles are obtained by intersecting P with planes of P3 not passing through v. In this way one obtains an ovoidal Laguerre plane of order pm . If the ovalO one starts off with is a conic, one obtains the Miquelian Laguerre plane of order pm . All known finite Laguerre planes of odd order are Miquelian. The internal incidence structure Ap at a point p of a Laguerre plane has the collection of all points not on the generator through p as point set and, as lines, all circles passing through p (without the point p) and all generators not passing through p. This is an affine plane, the derived affine plane at p. A circle K not passing through the distinguished point p induces an oval in the projective extension of the derived affine plane at p which intersects the line at infinity in the point corresponding to lines that come from generators of the Laguerre plane; inAp one has a parabolic curve. (The derived affine planes of the Miquelian Laguerre planes are Desarguesian and the parabolic curves are parabolae whose axes are the verticals, i.e., the lines that come from generators of the Laguerre plane.) A Laguerre plane can thus be described in one derived affine planeA by the lines ofA and a collection of parabolic curves. This planar description of a Laguerre plane, which is the most commonly used representation of a Laguerre plane, is then extended by the points of one generator where one has to adjoin a new point to each line and to each parabolic curve of the affine plane. It follows from [12] that every parabolic curve in a finite Desarguesian affine plane of odd order is in fact a parabola. Furthermore, using a simple counting argument, Chen and Kaerlein showed [2] that a finite Laguerre plane of odd order that admits a Desarguesian derivation is Miquelian.