q-Clan Geometries in Characteristic 2特征为2的q 氏族几何学

2007-10  

Birkhauser Verlag AG  

Cardinali, Ilaria,Payne, Stanley E.  

166  

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.

PreliminariesIntroductionFinite Generalized QuadranglesProlegomena1 q-Clans and Their Geometries　1.1 Anisotropism　1.2 q-Clans　1.3 Flocks of a Quadratic Cone　1.4 4-Gonal Families from q-Clans　1.5 Ovals in Ra　1.6 Herd Cover and Herd of Ovals　1.7 Herds of Ovals from q-Clans　1.8 Generalized Quadrangles from q-Clans　1.9 Spreads of PG(3,q) Associated with q-Clans2 The Fundamental Theorem　2.1 Grids and Affine Planes　2.2 The Fundamental Theorem　2.3 Aut(G)　2.4 Extension to 1/2-Normalized q-Clans　2.5 A Characterization of the q-Clan Kernel　2.6 Very Important Concept　2.7 The q-clan Cis, s F　2.8 The Induced Oval Stabilizers　2.9 Action of H on Generators of Cone K3　Aut(GQ(C))　3.1 General Remarks　3.2 An Involution of GQ(C)　3.3 The Automorphism Group of the Herd Cover 　3.4 The Magic Action of O'Keefe and Penttila 　3.5 The Automorphism Group of the Herd　3.6 The Groups Go, Go and G0　3.7 The Square-Bracket Function　3.8 A Cyclic Linear Collineation　3.9 Some Involutions　3.10 Some Semi-linear Collineations4 The Cyclic q-Clans　4.1 The Unified Construction of [COP03]　4.2 The Known Cyclic q-Clans　4.3 q-Clan Functions Via the Square Bracket　4.4 The Flip is a Collineation　4.5 The Main Isomorphism Theorem　4.6 The Unified Construction Gives Cyclic q-Clans 　4.7 Some Semi-linear Collineations　4.8 An Oval Stabilizer5　Applications to the Known Cyclic q-Clans　5.1 The Classical Examples: q=2e for e ≥1　5.2 The FTWKB Examples: q=2e with e Odd　5.3 The Subiaco Examples: q=2e, e≥4　5.4 The Adelaide Examples: q=2e with e Even6　The Subiaco Oval Stabilizers　6.1 Algebraic Plane Curves　6.2 The Action of Go on the Ra　6.3 The case e≡2 (mod 4)　6.4 The Case e≡10 (rood 20)　6.5 Subiaco Hyperovals: The Various Cases　6.6 O+(1,1) as an Algebraic Curve　6.7 The Case e≡O (rood 4)　6.8 The Case e Odd　6.9 The case e≡2 (rood 4)　6.10 Summary of Subiaco Oval Stabilizers7 The Adelaide Oval Stabilizers　7.1 The Adelaide Oval　7.2 A Polynomial Equation for the Adelaide Oval　7.3 Irreducibility of the Curve　7.4 The Complete Oval Stabilizer8 The Payne q-Clans9 Other Good StuffBibliographyIndex

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