Unitals in projective planes / Susan Barwick, Gary Ebert.

Format:

Book

Author/Creator:

Barwick, Susan.

Contributor:

Ebert, Gary.

Series:

Springer monographs in mathematics

Language:

English

Subjects (All):

Projective planes.

Physical Description:

xii, 193 pages : illustrations ; 24 cm.

Place of Publication:

[New York] : Springer, 2008.

Summary:

This clearly written text is the first book on unitals embedded in finite projective planes. Unitals are key structures in square order projective planes, and have connections with other structures in algebra. They provide a link between groups and geometries. There is a considerable number of research articles concerning unitals, and there also exist many open problems. This book is a thorough survey of the research literature on embedded unitals which collects this material in book form for the first time. The book is aimed at graduate students and researchers who want to learn about this topic without reading all the original articles.

The primary proof techniques used involve linear algebraic arguments, finite field arithmetic, some elementary number theory, and combinatorial enumeration. Some computer results not previously found in the literature also are mentioned in the text. The authors have included a comprehensive bibliography which will become an invaluable resource.

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.

Notes:

Includes bibliographical references (pages [177]-186) and indexes.

ISBN:

0387763643

9780387763644

OCLC:

251890769