Dimensional dual hyperovals in projective spaces and c.AGast-geometries
The two subjects of the title are studied, first independently and then by making them interact. Many questions arise from this interaction.In our intrinsic study of the first subject, we construct a new family of examples and define the notions of quotient and universality. For the second subject, we restrict the class c.AG* of circular extensions of dual affine spaces under the two geometrical conditions (LL) and (T) by showing that, apart from some extreme cases, every such rank three geometry can be erected into a rank 4 geometry. In particular, c.AG*n(q)-geometries satisfying (LL) and (T) with n ≥ 3 must have q even. We then deduce similar results for the first subject.
L · L*-geometries and Dn-buildings
Abstract A locally finite PG · PG∗ - geometry is a rank three incidence structure of points, lines and blocks , the block and dual point residues of which are {point,line}-systems of finite projective geometries and the line residues of which are generalized digons. We show that every locally finite thick PG · PG*-geometry satisfying two geometrical axioms (LL) and (T) is a truncation of a D n -building. We also strongly restrict geometries satisfying (LL) and (T) over the more general diagram L · PG*.
Some Infinite Families of Locally Partial Geometries Generalizing the Classical Laguerre Planes
Abstract Starting with a subgeometry Ω embedded in a β-dimensional projective space PG(β, q), β ⩾ 1, we construct inductively a series of rank n residually connected geometries Γ(n, β, Ω), n ⩾ β, by putting Γ(β, β, Ω) = Ω and extending Γ(n - 1, β, Ω) with a partial geometry.