Spreads Admitting Regular Elliptic Covers

We say that a spread S of PG(3, q) admits a regular elliptic cover if and only if S contains q−1 pairwise disjoint reguli (necessarily covering all but two fixed lines of S). Clearly, any André spread satisfies this condition. In this paper, we construct an infinite family of non-Andrd spreads admitting regular elliptic covers by replacing (q + 1)-nests of reguli in a regular spread. These are the only known non-André spreads to admit such a cover. The collineation groups of these spreads are also discussed in detail

Intriguing sets of strongly regular graphs and their related structures

In this paper we outline a technique for constructing directed strongly regular graphs by using strongly regular graphs having a "nice" family of intriguing sets. Further, we investigate such a construction method for rank three strongly regular graphs having at most $45$ vertices. Finally, several examples of intriguing sets of polar spaces are provided

On the Superabundance of Singular Varieties in Positive Characteristic

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not known what other examples there can be. Furthermore, before the work in this thesis, only a few examples of geproci nontrivial non-grid non-half-grids were known and there was no known way to generate more. Here, we use geometry in the positive characteristic setting to give new methods of producing geproci half-grids and non-half-grids. We also pick up work that had been done in 2017 by Solomon Akesseh, who had proven that there are no unexpected cubics in characteristic 3 with distinct points and gave examples involving infinitely near points based on quasi-elliptic fibrations in characteristic 2. Each quasi-elliptic fibration has a Dynkin diagram. Here, in contrast, for each possible Dynkin diagram for a quasi-elliptic fibration in characteristic 3, we give an example of the fibration but show it does not give rise to an unexpected cubic. [Equations Omitted] Adviser: Brian Harbourn

Regular parallelisms on PG(3,R) from generalized line stars: The oriented case

Using the Klein correspondence, regular parallelisms of PG(3,R) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3,R). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious

Classification of flocks of the quadratic cone in PG(3,64)

Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3,q) for q ≤ 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3,64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2,q), q even, and uses the prior classification of hyperovals in PG(2, 64)