Q-clan Geometries in Characteristic 2

A q-clan with q a power of 2 is equivalent to a certain generalized quadrangle with a family of subquadrangles each associated with an oval in the Desarguesian plane of order 2. It is also equivalent to a flock of a quadratic cone, and hence to a line-spread of 3-dimensional projective space and thus to a translation plane, and more. These geometric objects are tied together by the so-called Fundamental Theorem of q-Clan Geometry. The book gives a complete proof of this theorem, followed by a detailed study of the known examples. The collineation groups of the associated generalized quadrangles and the stabilizers of their associated ovals are worked out completely.

Preliminaries	p. ix
Introduction	p. ix
Finite Generalized Quadrangles	p. x
Prolegomena	p. xii
q-Clans and Their Geometries	p. 1
Anisotropism	p. 1
q-Clans	p. 2
Flocks of a Quadratic Cone	p. 2
4-Gonal Families from q-Clans	p. 4
Ovals in <$>\overline {{\cal R}}_{\alpha}<$>	p. 8
Herd Cover and Herd of Ovals	p. 9
Herds of Ovals from q-Clans	p. 11
Generalized Quadrangles from q-Clans	p. 12
Spreads of PG(3,q) Associated with q-Clans	p. 14
The Fundamental Theorem	p. 19
Grids and Affine Planes	p. 20
The Fundamental Theorem	p. 22
Aut (G^⊗)	p. 26
Extension to 1/2-Normalized q-Clans	p. 33
A Characterization of the q-Clan Kernel	p. 35
Very Important Concept	p. 38
The q-clan <$>{\cal C}^{i_s}<$>, s ∈ F	p. 38
The Induced Oval Stabilizers	p. 41
Action of <$>{\cal H}<$> on Generators of Cone K	p. 44
Aut(GQ(<$>{\cal C}<$>))	p. 47
General Remarks	p. 47
An Involution of GQ(<$>{\cal C}<$>)	p. 50
The Automorphism Group of the Herd Cover	p. 51
The Magic Action of O'Keefe and Penttila	p. 53
The Automorphism Group of the Herd	p. 59
The Groups <$>{\cal G}_0<$>, <$>G_0<$> and <$>\hat {G}_0<$>	p. 61
The Square-Bracket Function	p. 62
A Cyclic Linear Collineation	p. 63
Some Involutions	p. 65
Some Semi-linear Collineations	p. 66
The Cyclic q-Clans	p. 73
The Unified Construction of [COP03]	p. 73
The Known Cyclic q-Clans	p. 77
q-Clan Functions Via the Square Bracket	p. 78
The Flip is a Collineation	p. 80
The Main Isomorphism Theorem	p. 81
The Unified Construction Gives Cyclic q-Clans	p. 84
Some Semi-linear Collineations	p. 85
An Oval Stabilizer	p. 88
Applications to the Known Cyclic q-Clans	p. 91
The Classical Examples: q = 2^e for e ≥ 1	p. 91
The FTWKB Examples: q = 2^e with e Odd	p. 95
The Subiaco Examples: q = 2^e, e ≥ 4	p. 98
The Adelaide Examples: q = 2^e with e Even	p. 100
The Subiaco Oval Stabilizers	p. 101
Algebraic Plane Curves	p. 101
The Action of <$>{\cal G}_0<$> on the <$>{\cal R}_{\alpha}<$>	p. 104
The case e ≡ 2 (mod 4)	p. 106
The Case e ≡ 10 (mod 20)	p. 110
Subiaco Hyperovals: The Various Cases	p. 111
<$>{\cal O}_{(1.1)}^+<$> as an Algebraic Curve	p. 112
The Case e ≡ 0 (mod 4)	p. 116
The Case e Odd	p. 120
The case e ≡ 2 (mod 4)	p. 122
Summary of Subiaco Oval Stabilizers	p. 129
The Adelaide Oval Stabilizers	p. 133
The Adelaide Oval	p. 133
A Polynomial Equation for the Adelaide Oval	p. 133
Irreducibility of the Curve	p. 136
The Complete Oval Stabilizer	p. 137
The Payne q-Clans	p. 141
The Monomial q-Clans	p. 141
The Examples of Payne	p. 141
The Complete Payne Oval Stabilizers	p. 144
Other Good Stuff	p. 149
Spreads and Ovoids	p. 149
The Geometric Construction of J. A. Thas	p. 151
A Result of N. L. Johnson	p. 153
Translation (Hyper)Ovals and ¿-Flocks	p. 154
Monomial hyperovals	p. 156
Conclusion and open Problems	p. 157
Bibliography	p. 159
Index	p. 165
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Q-clan Geometries in Characteristic 2

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