Collineation groups of ovals and of ovoidal Laguerre planes (Q2563488)

An ovoidal Laguerre plane is the geometry of nontrivial plane sections of a cone \(C\) over an oval in 3-dimensional projective space. One obtains a miquelian Laguerre plane if the oval is a conic. For each plane of the projective space that intersects \(C\) in a circle its intersection with the cone yields an oval in this plane. The authors investigate the automorphism groups of ovoidal Laguerre planes and construct such planes with certain automorphism groups and ovals in infinite projective planes admitting only certain collineations. It is well-known that every automorphism of an ovoidal Laguerre plane is induced by a collineation of the surrounding projective space that leaves the cone invariant. In particular, the Laguerre plane admits all automorphisms that come from central collineations with centre being the vertex of the cone \(C\). The collection of all these automorphisms forms a normal subgroup \(\Delta\) in the full automorphism group \(\Sigma\) of the ovoidal Laguerre plane. Furthermore, \(\Sigma\) is the semidirect product of \(\Delta\) by the collineation group of the oval \(E\cap C\) in the plane \(E\), i.e., the stabilizer of \(E\cap C\) in the full collineation group of the desarguesian projective plane \(E\). Let \(P\) be an infinite denumerable projective plane. The authors show that \(P\) contains a rigid oval, i.e., an oval that is left invariant only by the identity of \(P\), and a rigid oval with a nucleus, i.e., all tangents pass through one point and every line through this point is a tangent to the oval. Furthermore, if \(\alpha\) is an involutory collineation of \(P\), then the authors construct an oval which is invariant under \(\alpha\) and also such an oval with nucleus. These results are achieved by carefully choosing points and lines that are to become the tangents of the oval in a process consisting of countably many steps, but finitely many points and lines are added at each step. Consequently, one obtains ovoidal Laguerre planes whose full automorphism group is \(\Delta\) and non-miquelian ovoidal Laguerre planes admitting involutory automorphisms which are not induced by projective collineations of the surrounding projective space, i.e., they do not come from linear maps of the corresponding 4-dimensional vector space. Furthermore, it is shown that for every involutory collineation \(\alpha\) of an infinite projective plane there exists a hyperoval invariant under \(\alpha\) such that the set of nonsecant lines is a prescribed set of lines of smaller cardinality than the cardinality of the projective plane. The authors further investigate certain ovoidal Laguerre planes of Dembowski type and completely determine the full automorphism groups of Laguerre planes over certain translation ovals.