Ovals and arcs in finite projective planes

Ph.D. thesis, University of Sussex at Brighton, U.K. Chapter 1 introduces some background material concerning finite projective planes, their arcs, conics, ovals and hyperovals. Chapters 2 and 3 are an update to a survey on constructions of ovals, Pascal ovals, abstract ovals in finite projective planes and their collineation groups. Chapter 4 deals with ovals in translation planes of odd order. In 1971 L. A. Rosati constructed a family of hyperbolic ovals in the regular nearfield plane π of dimension 2 over its centre. Three new results on Rosati ovals and their collineation groups are obtained: (i) Rosati ovals are pairwise equivalent under the action of the full collineation groups of π. (ii) Rosati ovals are of dihedral type. (iii) If a non-desarguesian translation plane of odd order contains a transitive hyperbolic oval of dihedral type, then the plane is a regular nearfield plane π of dimension 2 over its centre, and the oval is a Rosati oval. Chapter 5 is a thorough study of the 2-transitive ovals in the dual Lüneburg plane of even order. Some more evidence is given to support the conjecture dating back to the sixties that such ovals are the unique $2$-transitive ovals in finite non-desarguesian planes. Chapter 6 deals with the classification project of transitive ovals in projective planes of even order. A crucial open question is to determine projective planes of order $16$ that contain a transitive oval. The collineation group $G$ of such an oval has order at most $144$. To study the case where this bound is attained, a computational approach is employed. The new result is that the Lunelli-Sce-Hall oval in $\mathrm{PG}(2, 16)$ is the only such oval. Chapter 7 is about maximal arcs arising from conics in $\mathrm{PG}(2, q)$, with $q$ odd. A new proof depending on collineation groups is given for the classification theorem of maximal arcs of size $(q + 3)/2$ containing $(q + 1)/2$ points from a conic.