Q-CLAN GEOMETRIES IN CHARACTERISTIC 2
CONTENTS Q-CLANS AND THEIR GEOMETRIES ANISOTROPISM Q-CLANS FLOCKS OF A QUADRATIC CONE 4-GONAL FAMILIES FROM Q-CLANS OVALS IN R? HERD COVER AND HERD OF OVALS HERDS OF OVALS FROM Q-CLANS GENERALIZED QUADRANGLES FROM Q-CLANS SPREADS OF PG(3,Q) ASSOCIATED WITH Q-CLANS THE FUNDAMENTAL THEOREM GRIDS AND A?NE PLANES AUT(G?) EXTENSION TO 1/2-NORMALIZED Q-CLANS A CHARACTERIZATION OF THE Q-CLAN KERNEL VERY IMPORTANT CONCEPT THE Q-CLAN CIS, S ?F THE INDUCED OVAL STABILIZERS ACTION OF H ON GENERATORS OF CONE K AUT(GQ(C)) GENERAL REMARKS AN INVOLUTION OF GQ(C) THE AUTOMORPHISM GROUP OF THE HERD COVER THE MAGIC ACTION OF O’KEEFE AND PENTTILA THE AUTOMORPHISM GROUP OF THE HERD THE SQUARE-BRACKET FUNCTION A CYCLIC LINEAR COLLINEATION SOME INVOLUTIONS 3 10 SOME SEMI-LINEAR COLLINEATIONS THE CYCLIC Q-CLANS THE UNI?ED CONSTRUCTION OF [COP03] THE KNOWN CYCLIC Q-CLANS Q-CLAN FUNCTIONS VIA THE SQUARE BRACKET THE FLIP IS A COLLINEATION THE MAIN ISOMORPHISM THEOREM THE UNI?ED CONSTRUCTION GIVES CYCLIC Q-CLANS SOME SEMI-LINEAR COLLINEATIONS AN OVAL STABILIZER APPLICATIONS TO THE KNOWN CYCLIC Q-CLANS THE CLASSICAL EXAMPLES: Q = 2E FOR E ?1 THE FTWKB EXAMPLES: Q = 2E WITH E ODD THE SUBIACO EXAMPLES: Q = 2E, E ?4 THE ADELAIDE EXAMPLES: Q = 2E WITH E EVEN THE SUBIACO OVAL STABILIZERS ALGEBRAIC PLANE CURVES THE ACTION OF G0 ON THE R? THE CASE E ?2 (MOD 4) THE CASE E ?10 (MOD 20) SUBIACO HYPEROVALS: THE VARIOUS CASES O+ (1,1) AS AN ALGEBRAIC CURVE THE CASE E ?0 (MOD 4) THE CASE E ODD 6 10 SUMMARY OF SUBIACO OVAL STABILIZERS THE ADELAIDE OVAL STABILIZERS THE ADELAIDE OVAL A POLYNOMIAL EQUATION FOR THE ADELAIDE OVAL IRREDUCIBILITY OF THE CURVE THE COMPLETE OVAL STABILIZER THE PAYNE Q-CLANS THE MONOMIAL Q-CLANS THE EXAMPLES OF PAYNE THE COMPLETE PAYNE OVAL STABILIZERS OTHER GOOD STU? SPREADS AND OVOIDS THE GEOMETRIC CONSTRUCTION OF J A THAS A RESULT OF N L JOHNSON TRANSLATION (HYPER)OVALS AND ?-FLOCKS MONOMIAL HYPEROVALS CONCLUSION AND OPEN PROBLEMS