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Quadrals and Their Associated Subspaces

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Quadrals and Their Associated Subspaces School of Mathematical Sciences Discipline of Pure Mathematics Quadrals and their Associated Subspaces PhD Thesis Submitted May 2008 David Keith Butler Contents Abstract 5 Signed Statement 6 Acknowledgements 7 Introduction 8 0 Preliminaries 11 0.1Algebra.................................. 11 0.2ProjectiveSpaces............................. 18 0.3 Collineations and Polarities ....................... 24 0.4 Special sets of lines ............................ 29 1 Quadrals 36 1.1Ovals.................................... 36 1.2Ovoids................................... 41 1.3Cones................................... 48 1.4OvalandOvoidCones.......................... 58 1.5Quadrics.................................. 63 1.6Quadrals.................................. 83 2 2 Characterisations of the set of points of a quadral 87 2.1 Tallini Sets ................................ 87 2.2QuadraticSets.............................. 92 2.3Characterisationofquadralsbytheirplanesections.......... 97 3 Existing characterisations of the subspaces associated with quadrals100 3.1 Lines associated with quadrals in PG(2,q) ...............100 3.2 Lines associated with Quadrals in PG(n, q)...............103 4 Characterisation of the external lines of an oval cone in PG(3,q) 107 4.1 TheexternallinesofanovalconeinPG(3,q), q even.........107 4.2 TheexternallinesofanovalconeinPG(3,q), q odd..........114 5 Characterisation of subspaces associated with a parabolic quadric in PG(4,q) 124 5.1 The external lines of a parabolic quadric in PG(4,q), q even.....125 5.2 The secant planes of a parabolic quadric of PG(4,q)..........136 5.3 The tangents and generator lines of the non-singular quadric in PG(4,q), q odd....................................150 6 Characterisation of the external lines of quadrals of parabolic type in PG(n, q), q even 152 6.1 The external lines of a hyperoval cone in PG(n, q), q even.......153 6.2Theexternallinesofaquadralofparabolictype............160 3 7 The intersection of ovals and ovoids 166 7.1Theintersectionoftwoovals.......................167 7.2 Relationships Between Shared Points, Shared Tangents and Shared TangentPlanes..............................168 7.3 Intersections of elliptic quadrics. .....................171 7.4 Bounds on the intersection of ovoids. ..................172 7.5Ovoidalfibrations.............................175 7.6Theintersectionofovoidssharingalltangents.............179 7.7Theexternallinesorsecantsthattwoovoidsmayshare........184 Conclusion 188 Bibliography 190 4 Abstract This thesis concerns sets of points in the finite projective space PG(n, q)thatare combinatorially identical to quadrics. A quadric is the set of points of PG(n, q) whose coordinates satisfy a quadradic equation, and the term quadral is used in this thesis to mean a set of points with all the combinatorial properties of a quadric. Most of the thesis concerns the characterisation of certain sets of subspaces associ- ated with quadrals. Characterisations are proved for the external lines of an oval cone in PG(3,q), of a non-singular quadric in PG(4,q), q even, and of a large class of cones in PG(n, q), q even. Characterisations are also proved for the planes meeting the non-singular quadric of PG(4,q) in a non-singular conic, and for the tangents and generator lines of this quadric for q odd. The second part of the thesis is concerned with the intersection of ovoids of PG(3,q). A new bound is proved on the number of points two ovoids can share, and configu- rations of secants and external lines that two ovoids can share are determined. The structure of ovoidal fibrations is discussed, and this is used to prove new results on the intersection of two ovoids sharing all of their tangents. 5 Signed Statement This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being made available for loan and photocopying, subject to the provisions of the Copyright Act 1968. Signed: Date: 6 Acknowledgements I would like to acknowledge the reciept of a Faculty of Engineering, Computer and Mathematical Sciences Divisional Scholarship, without which it would not have been possible to undertake this research degree. I must thank my supervisors Susan Barwick and Rey Casse for their support, guid- ance, criticism, encouragement and chocolate. Sincerest thanks also go to Stephanie Lord, Jan Sutton and Di Parrish for answering all sorts of questions and solving all sorts of problems with a smile. I would also like to thank the other postgraduate and honours students, who have made the experience not only bearable but actually enjoyable. Thank you to ev- eryone for playing croquet, testing new games, listening to my latest proof and just being there with me on the journey. I particularly want to thank my good friends Ric Green, David Roberts, Ray Vozzo and Jono Tuke for being my good friends. Finally, I need to thank my precious wife Catherine. How could I possibly live without you my darling, letalone finish a PhD thesis? You supported me right from the first idea that a PhD might be a possibility, all the way along the winding road to this the last long wait. These three years have really been worth it. Not because of the reward I’ll get from all this hard work, but because I know even more surely that you’ll always be with me, no matter what happens. 7 Introduction The tension between algebra and combinatorics is central to the study of finite geometry. The finite projective space PG(n, q) itself (for n>2) can be equally well defined algebraically using the underlying field GF(q) or combinatorially using the axioms of incidence. This connection between algebra and combinatorics in the very definition of projective space leads to the belief that similar connections should exist foreachobjectcontainedinPG(n, q). That is, an object with an algebraic definition is expected to have an equally valid combinatorial definition; and vice versa an object with a combinatorial definition is expected to have a simple algebraic description. The simplest algebraic description is a linear equation, and the points satisfying linear equations are exactly the subspaces. The next simplest algebraic description is a quadratic equation and the set of points defined by a quadratic equation is called a quadric. The connection between algebra and combinatorics leads to the expectation that quadrics should also have a valid combinatorial definition. That is, it should be possible to show that if a set of points has some key combinatorial properties of a quadric, then it must in fact be a quadric. Such a result is called a characterisation result – the quadrics would be characterised by these key properties. Unfortunately, it is not possible to characterise certain quadrics using only combi- natorial properties. In certain cases, there exist objects with all the combinatorial properties of quadrics, but which are not quadrics. That is, their points do not all satisfy a quadratic equation. In this thesis, the term quadral is used to mean a set of points combinatorially identical to a quadric. A large part of this thesis is concerned with characterisation, but not characterisation of the quadrals themselves. Instead the focus is on characterising the families of subspaces associated with the quadral. One of the key properties of quadrals is that 8 subspaces can only meet a quadral in certain ways. The subspaces of PG(n, q)can be grouped into families based on how they meet a particular quadral, and then each family has its own combinatorial properties. For example, consider the lines that contain no point of the quadral Q –theexternal lines of Q. This set of lines has many combinatorial properties. Given a set of lines L with some of these properties, the aim is to prove that L is in fact the family of external lines to a quadral like Q. In a sense, the family of lines is used to reconstruct the quadral Q. This leads to the remaining part of the thesis. Given that the external lines of Q can be used to reconstruct Q, it is natural to ask how much information is required to fully determine Q. For example, how many external lines are required to fully determine Q? Another way to ask this is to ask how many external lines two quadrals may share. The second part of the thesis concerns this type of problem for a particular family of quadrals – the ovoids of PG(3,q). Outline of Thesis The thesis begins with the necessary background information. Chapter 0 contains the basic definitions and results of projective geometry that are needed to discuss quadrals. Chapter 1 describes quadrals, which are the objects of interest in this thesis. The definitions and important properties of quadrics, ovals, ovoids and their cones are included here. Chapter 2 discusses the characterisation of quadrals. Several results are stated characterising quadrals by their intersections with lines and planes, including the characterisations of Tallini, Buekenhout and Thas. These characterisations will be used in later chapters to prove characterisations of the families of subspaces associated with quadrals. Chapter 3 discusses the existing characterisations of the families of subspaces asso- ciated with quadrals. Of particular note are those characterising the external lines of an ovoid and of a hyperbolic quadric in PG(3,q). The next three chapters present original results characterising certain families of subspaces associated with quadrals. Chapter 4 presents characterisations of the 9 external lines of an oval cone in PG(3,q)forq odd and q even. Chapter 5 presents characterisations of the planes meeting a parabolic quadric of PG(4,q) in a non- singular conic, and of the external lines of this quadric when q is even. Chapter 6 presents characterisations of the external lines of an oval cone in PG(n, q), and also the external lines of a cone over any parabolic quadric.
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