q-Clan Geometries in Characteristic 2 (eBook)

Contents 6
Preliminaries 9
Introduction 9
Finite Generalized Quadrangles 10
Prolegomena 12
q-Clans and Their Geometries 15
1.1 Anisotropism 15
1.2 q-Clans 16
1.3 Flocks of a Quadratic Cone 16
1.4 4-Gonal Families from q-Clans 18
1.5 Ovals in Ra 22
1.6 Herd Cover and Herd of Ovals 23
1.7 Herds of Ovals from q-Clans 25
1.8 Generalized Quadrangles from q-Clans 26
1.9 Spreads of PG(3, q) Associated with q-Clans 28
The Fundamental Theorem 32
2.1 Grids and Affne Planes 33
2.2 The Fundamental Theorem 35
2.3 Aut(G.) 39
2.4 Extension to 1/2-Normalized q-Clans 46
2.5 A Characterization of the q-Clan Kernel 48
2.6 Very Important Concept 51
2.7 The q-clan Cis, s . F 51
2.8 The Induced Oval Stabilizers 54
2.9 Action of H on Generators of Cone K 57
Aut(GQ(C)) 59
3.1 General Remarks 59
3.2 An involution of GQ(C) 62
3.3 The Automorphism Group of the Herd Cover 63
3.4 The Magic Action of O’Keefe and Penttila 65
3.5 The Automorphism Group of the Herd 71
3.6 The Groups G0, G0 and G0 73
3.7 The Square-Bracket Function 74
3.8 A Cyclic Linear Collineation 75
3.9 Some Involutions 77
3.10 Some Semi-linear Collineations 78
The Cyclic q-Clans 85
4.1 The Uni.ed Construction of [COP03] 85
4.2 The Known Cyclic q-Clans 89
4.3 q-Clan Functions Via the Square Bracket 90
4.4 The Flip is a Collineation 92
4.5 The Main Isomorphism Theorem 93
4.6 The Uni.ed Construction Gives Cyclic q-Clans 96
4.7 Some Semi-linear Collineations 97
4.8 An Oval Stabilizer 100
Applications to the Known Cyclic q-Clans 103
5.1 The Classical Examples: q = 2e for e = 1 103
5.2 The FTWKB Examples: q = 2e with e Odd 107
5.3 The Subiaco Examples: q = 2e, e = 4 110
5.4 The Adelaide Examples: q = 2e with e Even 112
The Subiaco Oval Stabilizers 113
6.1 Algebraic Plane Curves 113
6.2 The Action of G0 on the Ra 116
6.3 The Case e = 2 (mod 4) 118
6.4 The Case e = 10 (mod 20) 122
6.5 Subiaco Hyperovals: The Various Cases 123
6.6 O+(1,1) as an Algebraic Curve 124
6.7 The Case e = 0 (mod 4) 128
6.8 The Case e Odd 132
6.9 The case e = 2 (mod 4) 134
6.10 Summary of Subiaco Oval Stabilizers 141
The Adelaide Oval Stabilizers 144
7.1 The Adelaide Oval 144
7.2 A Polynomial Equation for the Adelaide Oval 144
7.3 Irreducibility of the Curve 147
7.4 The Complete Oval Stabilizer 148
The Payne q-Clans 151
8.1 The Monomial q-Clans 151
8.2 The Examples of Payne 151
8.3 The Complete Payne Oval Stabilizers 154
Other Good Stuff 159
9.1 Spreads and Ovoids 159
9.2 The Geometric Construction of J. A. Thas 161
9.3 A Result of N. L. Johnson 163
9.4 Translation (Hyper)Ovals and a-Flocks 164
9.5 Monomial hyperovals 166
9.6 Conclusion and open Problems 167
Bibliography 168
Index 174

Preliminaries (p. ix-x)

Introduction

This memoir is a thoroughly revised and updated version of the Subiaco Notebook, which since early 1998 has been available on the web page of the senior author at http://www-math.cudenver.edu/¡«spayne/.

Our goal is to give a fairly complete and nearly self-contained treatment of the known (infinite families of) generalized quadrangles arising from q-clans, i.e., flocks of quadratic cones in PG(3, q), with q = 2e. Our main interest is in the construction of the generalized quadrangles, a determination of the associated ovals, and then a complete determination of the groups of automorphisms of these objects. A great deal of general theoretical material of related interest has been omitted. However, we hope that the reader will find the treatment here to be coherent and complete as a treatment of one major part of the theory of flock generalized quadrangles.

Since the appearance of the Subiaco Notebook the Adelaide q-clans have been discovered, generalizing the few examples of "cyclic" q-clans found first by computer (see [PPR97]). The revised treatment given here of the cyclic q-clans, which is a slight improvement of that given in [COP03], allows much of the onerous computation in the Subiaco Notebook to be avoided while at the same time allowing a more unified approach to the general subject. However, a great deal of computation is still unavoidable.

Most of the work done on the Adelaide examples, especially our study of the Adelaide ovals, and major steps in the clarification of the connection between the so-called Magic Action of O’Keefe and Penttila (see [OP02]) and the Fundamental Theorem of q-clan geometry, took place while the senior author was a visiting research professor at the Universities of Naples, Italy, and Ghent, Belgium, during the winter and spring of the year 2002. This was made possible by a semester-long sabbatical provided by the author’s home institution, the University of Colorado at Denver.

During the two months he spent in Italy, at the invitation of Professor Dr. Guglielmo Lunardon (with the collaboration of Prof. Laura Bader), he received generous financial support from the GNSAGA and the University of Naples, along with a great deal of personal support from his colleagues there. Also, much of the material on the cyclic q-clans derives from the reports [Pa02a] and [Pa02b] and has appeared in [CP03].

During his two months in Belgium, at the invitation of Prof. Dr. Joseph A. Thas, he was generously supported by the Research Group in Incidence Geometry at Ghent University. As always, it was a truly great pleasure to work in the stimulating and friendly atmosphere provided by his colleagues there. All the material on the Adelaide ovals was adapted from [PT05].

It was in Naples during the trimester February-March 2002 that the second author became familiar with the Four Lectures in Naples [Pa02a] and the idea of working with the senior author to complete the present memoir first occured to us.