Unitals in Projective Planes (eBook)

Susan Barwick, Gary Ebert (Autoren)

eBook Download: PDF

2009 | 2008
XII, 196 Seiten
Springer New York (Verlag)
978-0-387-76366-8 (ISBN)

Infos

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.

Inhalt

Preface 6
Contents 9
1 Preliminaries 11
Affine and Projective Geometries 11
Finite Fields 18
Quadrics in Low Dimensions 19
Ovals and Ovoids 22
Some Linear Algebra 23
2 Hermitian Curves and Unitals 31
Nondegenerate Hermitian Curves 31
Degenerate Hermitian Curves and Baer Sublines 34
Unitals 37
3 Translation Planes 42
Translation Planes 42
Derivation 43
Spreads 45
The Bruck-Bose Representation 50
The Bruck-Bose Construction 50
Baer Subplanes and Baer Sublines in Bruck-Bose 52
Derivation in Bruck-Bose 61
Coordinates in Bruck-Bose 62
4 Unitals Embedded in Desarguesian Planes 67
Buekenhout Constructions 67
Unitals Embedded in PG(2,q2) 74
The Odd Characteristic Case 74
The Even Characteristic Case 87
5 Unitals Embedded in Non-Desarguesian Planes 96
Unitals in Hall Planes 96
Unitals in Semifield Planes 104
Unitals in Nearfield Planes 108
Unitals Embedded in Nontranslation Planes 110
Figueroa Plane 110
Hughes Plane 112
6 Combinatorial Questions and Associated Configurations 116
Intersection Problems 116
Spreads and Packings 124
Related Combinatorial Structures 128
Inversive Planes 128
Arcs 130
Unitals and Codes 134
7 Characterization Results 139
Characterizations of Unitals via Baer Sublines 139
Proofs of Results from Section 7.1 142
Other Configurational Characterizations 154
Tallini Scafati Characterizations 154
Characterizations Using Feet 159
Characterizations Using O'Nan Configurations 167
Characterizations Using the Quadratic Extension PG(4,q2) 170
The Bose Representation of PG(2,q2) in PG(5,q) 170
Group Theoretic Characterizations 172
8 Open Problems 173
A Nomenclature of Unitals 176
B Group Theoretic Characterizations of Unitals 178
References 182
Notation Index 192
Index 193

Produkt-Details

• Erscheint lt. Verlag: 3.4.2009
• Reihe/Serie: Springer Monographs in Mathematics
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: XII, 196 p. 29 illus.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: Algebra • classification • combinatorics • Field • linear algebra • Number Theory
• ISBN-10: 0-387-76366-X / 038776366X
• ISBN-13: 978-0-387-76366-8 / 9780387763668