Unitals in projective planes revisited

Abstract

This thesis revisits the topic of unitals in finite projective planes. A unital U in a projective plane of order q2 is a set of q3 + 1 points, such that every line meets U in one or q + 1 points. Unitals are an important class of point-set in finite projective planes, whose combinatorial and algebraic properties have been the subject of considerable study. In this work, we summarise, revise, and extend contemporary research on unitals. Chapter 1 covers the necessary prerequisites to study unitals and related objects in finite geometry. In Chapter 2, we focus on Buekenhout-Tits unitals and answer some open problems regarding their equivalence, stabilisers and feet. The results presented in Chapter 2 are also available in a preprint paper [22]. Following this, Chapter 3 summarises recent results on Buekenhout- Metz unitals, and presents a small result on the intersection of ovoidal-Buekenhout-Metz unitals and Buekenhout-Metz unitals. Chapter 4 highlights Kestenband arcs and their relationship to Hermitian unitals, and makes explicit a proof of their equivalence. Finally in Chapter 5, we review our understanding of Figueroa planes. Beyond describing ovals and unitals in Figueroa planes, we also suggest generalisations of their constructions to semi-ovals