Scattered spaces with respect to spreads, and eggs in finite projective spaces (Q2761077)

This dissertation is the report of the research that the author performed during his years as a PhD student. It consists of two main parts. NEWLINENEWLINENEWLINEThe first main part discusses scattered subspaces with respect to spreads in finite projective spaces. A spread in a projective space \(\Pi\) is a partition of \(\Pi\) in subspaces of the same dimension. A subspace is called scattered with respect to a spread if it intersects every element of that spread in at most one point. An upper bound on the dimension of a scattered subspace is derived. A lower bound is derived if the scattered subspace is not properly contained in another scattered subspace. The obtained bounds can be improved in the case that the spread is Desarguesian. NEWLINENEWLINENEWLINEAlso the connection with two-intersection sets and blocking sets is explained. A set \(X\) of points in a projective space \(\Pi\) is called a two-intersection set with respect to hyperplanes, if \(|X \cap \pi|\) takes exactly two values if \(\pi\) ranges over all hyperplanes of \(\Pi\). An \(s\)-fold blocking set with respect to \(k\)-dimensional subspaces in PG\((n,q)\) is a set of points, at least \(s\) on every \(k\)-dimensional subspace of PG\((n,q)\). NEWLINENEWLINENEWLINEThe second main part studies so-called eggs in finite projective spaces, these are certain sets of subspaces in projective spaces. The theory of eggs is equivalent with the theory of the so-called translation generalized quadrangles, see [\textit{S. E. Payne} and \textit{J. A. Thas}, Finite generalized quadrangles (Research Notes in Mathematics 110, Pitman, Boston) (1984; Zbl 0551.05027)]. NEWLINENEWLINENEWLINEThere is also a connection with the theory of flocks of quadratic cones. Such a flock is defined as a partition of the cone minus its vertex in irreducible conics. NEWLINENEWLINENEWLINEMany calculations performed in the report use a certain model for the egg. If the egg is a so-called good egg in PG\((4n-1,q)\), then the corresponding translation generalized quadrangle contains many subquadrangles of type \(Q(4,q^n)\). Every point outside such a subquadrangle then defines a subtended ovoid of \(Q(4,q^n)\). These ovoids are studied. A new characterization of the eggs of Kantor type is given, as well as an important result towards the classification of the so-called semifield flocks. Also all eggs in the seven-dimensional projective space over the field of order 2 are determined.