Unitals in projective planes / Susan Barwick, Gary Ebert.
"This book is a monograph on unitals embedded in finite projective planes. Unitals are key structures in projective planes, and have connections with other structures in algebra. They play a significant role in the classification of finite planes and provide a link between groups and geometries...
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| Main Authors: | , |
|---|---|
| Format: | Ebook |
| Language: | English |
| Published: |
[New York] :
Springer,
2008.
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| Series: | Springer monographs in mathematics.
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| Subjects: | |
| Online Access: | Springer eBooks |
Table of Contents:
- 1.1 Affine and Projective Geometries 1
- 1.2 Finite Fields 8
- 1.3 Quadrics in Low Dimensions 9
- 1.4 Ovals and Ovoids 12
- 1.5 Some Linear Algebra 13
- 2 Hermitian Curves and Unitals 21
- 2.1 Nondegenerate Hermitian Curves 21
- 2.2 Degenerate Hermitian Curves and Baer Sublines 24
- 2.3 Unitals 27
- 3 Translation Planes 33
- 3.1 Translation Planes 33
- 3.2 Derivation 34
- 3.3 Spreads 36
- 3.4 The Bruck-Bose Representation 41
- 3.4.1 The Bruck-Bose Construction 41
- 3.4.2 Baer Subplanes and Baer Sublines in Bruck-Bose 43
- 3.4.3 Derivation in Bruck-Bose 52
- 3.4.4 Coordinates in Bruck-Bose 53
- 4 Unitals Embedded in Desarguesian Planes 59
- 4.1 Buekenhout Constructions 59
- 4.2 Unitals Embedded in PG(2, q[superscript 2]) 66
- 4.2.1 The Odd Characteristic Case 66
- 4.2.2 The Even Characteristic Case 79
- 5 Unitals Embedded in Non-Desarguesian Planes 89
- 5.1 Unitals in Hall Planes 89
- 5.2 Unitals in Semifield Planes 97
- 5.3 Unitals in Nearfield Planes 101
- 5.4 Unitals Embedded in Nontranslation Planes 103
- 5.4.1 Figueroa Plane 103
- 5.4.2 Hughes Plane 105
- 6 Combinatorial Questions and Associated Configurations 109
- 6.1 Intersection Problems 109
- 6.2 Spreads and Packings 117
- 6.3 Related Combinatorial Structures 121
- 6.3.1 Inversive Planes 121
- 6.3.2 Arcs 123
- 6.4 Unitals and Codes 127
- 7 Characterization Results 133
- 7.1 Characterizations of Unitals via Baer Sublines 133
- 7.2 Proofs of Results from Section 7.1 136
- 7.3 Other Configurational Characterizations 148
- 7.3.1 Tallini Scafati Characterizations 148
- 7.3.2 Characterizations Using Feet 153
- 7.3.3 Characterizations Using O'Nan Configurations 161
- 7.4 Characterizations Using the Quadratic Extension PG(4, q[superscript 2]) 164
- 7.5 The Bose Representation of PG(2, q[superscript 2]) in PG(5, q) 164
- 7.6 Group Theoretic Characterizations 166
- 8 Open Problems 167
- A Nomenclature of Unitals 171
- B Group Theoretic Characterizations of Unitals 173.