On Buekenhout-Metz unitals of even order

AbstractThe even order Buekenhout-Metz unitals are enumerated (up to projective equivalence) and their inherited collineation groups are computed. They are shown to be self-dual as designs, and certain related designs are also constructed

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

A sporadic simple group of B. Fischer of order 64, 561, 751, 654, 400

In this thesis we study a sporadic simple group M ( 22 ) of order 64,561,751, 6 54,400 = 2 17 .39.5 2.7. .1.13 . defined by 13. Fischer [3]. In chapter I most of the material needed from other sources is presented. For example, in §1 we outline results from the paper "Finite groups generated by 3-trallspositions" recently written by B. Fischer [3]. * In chapter II we deal with the calculation of the character tables of PSU(5,2), P8U(6,2), PSΩ+(6,3) PSΩ(7,3) and M(22). In chapter III a characterization of M(22) by the structure of the centralizer of' one of' its involutions is given. The involution is not the involution central in the Sylow 2-subgroup but is a 3-transposition

Classification of Non-Singular Cubic Surfaces up to e-invariants

In this thesis, we use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3, q) from 6 points in general position in PG(2, q) for q = 17, 19, 23, 29, 31. We classify the cubic surfaces with twenty-seven lines in three dimensions (up to e- invariants) by introducing computational and geometrical procedures for the classi- fication. All elliptic and hyperbolic lines on a non-singular cubic surface in PG(3, q) for q = 17, 19, 23, 29, 31 are calculated. We define an operation on triples of lines on a non-singular cubic surface with 27 lines which help us to determine the exact value of the number of Eckardt point on a cubic surface. Moreover, we discuss the irreducibil- ity of classes of smooth cubic surfaces in PG(19, C), and we give the corresponding codimension of each class as a subvariety of PG(19, C)

Weighted (k,n)-arcs in the projective plane of order nine

In this study the presence of (k,n;f)-arcs of two characters (m,n) in the Projective Plane of order 9 was investigated.The example of (85,13;f)-arcs of type (10,13) with the number of points of weight 0 is 6 was discussed. It was proved that there is no (85,13;f)-arc when the points of weight 0 form a (6,3)-arcs in PG(2,9). It was proved that there are two examples of (85,13;f)-arcs of type (10,13) when the points of weight 0 form a 6-arc which lies on a conic.The example of (81,12;f)-arcs of type (9,12) with the number of points of weight 0 is 10 was discussed. It was proved that there is no (81,12;f)-arc when the points of weight 0 form a a conic or a (10,3)-arc in PG(2,9).<p

The cubic surfaces with twenty-seven lines over finite fields

In this thesis, we classify the cubic surfaces with twenty-seven lines in three dimensional projective space over small finite fields. We use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3; q) from 6-arcs not on a conic in PG(2; q). We introduce computational and geometrical procedures for the classification of cubic surfaces over the finite field Fq. The performance of the algorithms is illustrated by the example of cubic surfaces over F13, F17 and F19

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4