A survey on semiovals

A semioval in a finite projective plane is a non-empty pointset S with the property that for every point in $S$ there exists a unique line t_P such that $S \cap t_P = {P}$. This line is called the tangent to S at P. Semiovals arise in several parts of finite geometries: as absolute points of a polarity (ovals, unitals), as special minimal blocking sets (vertexless triangle), in connection with cryptography (determining sets). We survey the results on semiovals and give some new proofs

2-semiarcs in PG(2, q), q <= 13

A 2-semiarc is a pointset S-2 with the property that the number of tangent lines to S-2 at each of its points is two. Using some theoretical results and computer aided search, the complete classification of 2-semiarcs in PG(2, q) is given for q <= 7, the spectrum of their sizes is determined for q <= 9, and some results about the existence are proven for q = 11 and q = 13. For several sizes of 2-semiarcs in PG(2, q), q <= 7, classification results have been obtained by theoretical proofs

Unitals in projective planes revisited

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