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.
A Family of d -dimensional Dual Hyperovals in PG(2d+1, 2)
With the affine part of an oval we associate a family of d -subspaces ofPG (2 d+ 1, 2) which can be thought of as a higher dimensional analogue of a hyperoval. The isomorphisms among such families together with their automorphisms are determined when they come from translation ovals.
Gluing two affine spaces.
A construction is described in [2] by which, given two or more geometries of the same rank n, each equipped with a suitable parallelism giving rise to the same geometry at infinity, we can glue them together along their geometries at infinity, thus obtaininig a new geometry of rank n+ k− 1, k being the number of geometries we glue. In this paper we will examine a special case of that construction, namely the gluing of two affine spaces.