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.
Author(s): Ilaria Cardinali, Stanley E. Payne
Edition: 1
Year: 2007
Language: English
Pages: 181
Cover......Page 1
Frontiers in Mathematics......Page 3
q-Clan Geometries in Characteristic 2......Page 4
Copyright - ISBN: 3764385073......Page 5
Contents......Page 6
Introduction......Page 10
Finite Generalized Quadrangles......Page 11
Prolegomena......Page 13
1.1 Anisotropism......Page 16
1.3 Flocks of a Quadratic Cone......Page 17
1.4 4-Gonal Families from q-Clans......Page 19
1.5 Ovals in \bar R[sub(α)]......Page 23
1.6 Herd Cover and Herd of Ovals......Page 24
1.7 Herds of Ovals from q-Clans......Page 26
1.8 Generalized Quadrangles from q-Clans......Page 27
1.9 Spreads of PG(3,q) Associated with q-Clans......Page 29
2. The Fundamental Theorem......Page 34
2.1 Grids and Affine Planes......Page 35
2.2 The Fundamental Theorem......Page 37
2.3 Aut(G[sup(⊗)])......Page 41
2.4 Extension to 1/2-Normalized q-Clans......Page 48
2.5 A Characterization of the q-Clan Kernel......Page 50
2.7 The q-clan C[sup(i_s)] , s ∈F......Page 53
2.8 The Induced Oval Stabilizers......Page 56
2.9 Action of H on Generators of Cone K......Page 59
3.1 General Remarks......Page 62
3.2 An Involution of GQ(C)......Page 65
3.3 The Automorphism Group of the Herd Cover......Page 66
3.4 The Magic Action of O'Keefe and Penttila......Page 68
3.5 The Automorphism Group of the Herd......Page 74
3.6 The Groups G[sub(0)], G[sub(0)] and G[sub(0)]......Page 76
3.7 The Square-Bracket Function......Page 77
3.8 A Cyclic Linear Collineation......Page 78
3.9 Some Involutions......Page 80
3.10 Some Semi-linear Collineations......Page 81
4.1 The Unified Construction of [COP03]......Page 88
4.2 The Known Cyclic q-Clans......Page 92
4.3 q-Clan Functions Via the Square Bracket......Page 93
4.4 The Flip is a Collineation......Page 95
4.5 The Main Isomorphism Theorem......Page 96
4.6 The Unified Construction Gives Cyclic q-Clans......Page 99
4.7 Some Semi-linear Collineations......Page 100
4.8 An Oval Stabilizer......Page 103
5.1 The Classical Examples: q = 2[sup(e)] for e ≥ 1......Page 106
5.2 The FTWKB Examples: q = 2[sup(e)] with e Odd......Page 110
5.3 The Subiaco Examples: q = 2[sup(e)], e ≥ 4......Page 113
5.4 The Adelaide Examples: q = 2[sup(e)] with e Even......Page 115
6.1 Algebraic Plane Curves......Page 116
6.2 The Action of G[sub(0)] on the R[sub(α)]......Page 119
6.3 The case e ≡ 2 (mod 4)......Page 121
6.4 The Case e ≡ 10 (mod 20)......Page 125
6.5 Subiaco Hyperovals: The Various Cases......Page 126
6.6 O[sup(+)][sub((1,1))] as an Algebraic Curve......Page 127
6.7 The Case e ≡ 0 (mod 4)......Page 131
6.8 The Case e Odd......Page 135
6.9 The case e ≡ 2 (mod 4)......Page 137
6.10 Summary of Subiaco Oval Stabilizers......Page 144
7.2 A Polynomial Equation for the Adelaide Oval......Page 148
7.3 Irreducibility of the Curve......Page 151
7.4 The Complete Oval Stabilizer......Page 152
8.2 The Examples of Payne......Page 156
8.3 The Complete Payne Oval Stabilizers......Page 159
9.1 Spreads and Ovoids......Page 164
9.2 The Geometric Construction of J. A. Thas......Page 166
9.3 A Result of N. L. Johnson......Page 168
9.4 Translation (Hyper)Ovals and α-Flocks......Page 169
9.5 Monomial hyperovals......Page 171
9.6 Conclusion and open Problems......Page 172
Bibliography......Page 174
S......Page 180
