Barwick / Ebert Unitals in Projective Planes

1. Auflage 2009
ISBN: 978-0-387-76366-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 196 Seiten

Reihe: Springer Monographs in Mathematics

This book is a monograph on unitals embedded in ?nite projective planes. Unitals are an interesting structure found in square order projective planes, and numerous research articles constructing and discussing these structures have appeared in print. More importantly, there still are many open pr- lems, and this remains a fruitful area for Ph.D. dissertations. Unitals play an important role in ?nite geometry as well as in related areas of mathematics. For example, unitals play a parallel role to Baer s- planes when considering extreme values for the size of a blocking set in a square order projective plane (see Section 2.3). Moreover, unitals meet the upper bound for the number of absolute points of any polarity in a square order projective plane (see Section 1.5). From an applications point of view, the linear codes arising from unitals have excellent technical properties (see 2 Section 6.4). The automorphism group of the classical unitalH =H(2,q ) is 2-transitive on the points ofH, and so unitals are of interest in group theory. In the ?eld of algebraic geometry over ?nite ?elds,H is a maximal curve that contains the largest number of F -rational points with respect to its genus, 2 q as established by the Hasse-Weil bound.

Autoren/Hrsg.

Barwick, Susan

Ebert, Gary

Inhaltsverzeichnis

1;Preface;6
2;Contents;9
3;1 Preliminaries;11
3.1;Affine and Projective Geometries;11
3.2;Finite Fields;18
3.3;Quadrics in Low Dimensions;19
3.4;Ovals and Ovoids;22
3.5;Some Linear Algebra;23
4;2 Hermitian Curves and Unitals;31
4.1;Nondegenerate Hermitian Curves;31
4.2;Degenerate Hermitian Curves and Baer Sublines;34
4.3;Unitals;37
5;3 Translation Planes;42
5.1;Translation Planes;42
5.2;Derivation;43
5.3;Spreads;45
5.4;The Bruck-Bose Representation;50
5.4.1;The Bruck-Bose Construction;50
5.4.2;Baer Subplanes and Baer Sublines in Bruck-Bose;52
5.4.3;Derivation in Bruck-Bose;61
5.4.4;Coordinates in Bruck-Bose;62
6;4 Unitals Embedded in Desarguesian Planes;67
6.1;Buekenhout Constructions;67
6.2;Unitals Embedded in PG(2,q2);74
6.2.1;The Odd Characteristic Case;74
6.2.2;The Even Characteristic Case;87
7;5 Unitals Embedded in Non-Desarguesian Planes;96
7.1;Unitals in Hall Planes;96
7.2;Unitals in Semifield Planes;104
7.3;Unitals in Nearfield Planes;108
7.4;Unitals Embedded in Nontranslation Planes;110
7.4.1;Figueroa Plane;110
7.4.2;Hughes Plane;112
8;6 Combinatorial Questions and Associated Configurations;116
8.1;Intersection Problems;116
8.2;Spreads and Packings;124
8.3;Related Combinatorial Structures;128
8.3.1;Inversive Planes;128
8.3.2;Arcs;130
8.4;Unitals and Codes;134
9;7 Characterization Results;139
9.1;Characterizations of Unitals via Baer Sublines;139
9.2;Proofs of Results from Section 7.1;142
9.3;Other Configurational Characterizations;154
9.3.1;Tallini Scafati Characterizations;154
9.3.2;Characterizations Using Feet;159
9.3.3;Characterizations Using O'Nan Configurations;167
9.4;Characterizations Using the Quadratic Extension PG(4,q2);170
9.5;The Bose Representation of PG(2,q2) in PG(5,q);170
9.6;Group Theoretic Characterizations;172
10;8 Open Problems;173
11;A Nomenclature of Unitals;176
12;B Group Theoretic Characterizations of Unitals;178
13;References;182
14;Notation Index;192
15;Index;193