Cardinali / Payne q-Clan Geometries in Characteristic 2

2007
ISBN: 978-3-7643-8508-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 166 Seiten, eBook

Reihe: Frontiers in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Cardinali, Ilaria

Payne, Stanley E.

Inhaltsverzeichnis

q-Clans and Their Geometries.- The Fundamental Theorem.- Aut(GQ(C)).- The Cyclic q-Clans.- Applications to the Known Cyclic q-Clans.- The Subiaco Oval Stabilizers.- The Adelaide Oval Stabilizers.- The Payne q-Clans.- Other Good Stuff.

Leseproben

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.