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Characterising Substructures of Finite Projective Spaces

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Characterising Substructures of Finite Projective Spaces Vrije Universiteit Brussel - Faculty of Science and Bio-engineering Sciences Ghent University - Faculty of Science Characterising substructures of finite projective spaces Sara Rottey Promotors: Prof. Dr. Philippe Cara Prof. Dr. Leo Storme Dr. Geertrui Van de Voorde 2015 Graduation thesis submitted in fulfillment of the requirements for the degree of Doctor in Science: Mathematics On a dit souvent que la g´eom´etrie est l’art de bien raisonner sur des figures mal faites. – Henri Poincar´e Dankwoord De voorbije vier jaar heb ik het geluk gehad mij te mogen verdiepen in de wondere wereld van de eindige meetkunde. Ik kreeg bovendien de kans om veel interessante mensen te ontmoeten en verschillende delen van de wereld te zien. Het spreekt voor zich dat dit proefschrift niet volledig zou zijn zonder de vele mensen die me onderweg geholpen hebben te bedanken voor hun steun en aanmoedigingen. Bedankt Geertrui, om me in te wijden in de geheimen van de veldreductie. Samen- werken met jou was puur plezier. Bedankt voor je kritische blik, verfrissende eerlijkheid en eeuwig geduld als ik begon te ratelen. Zonder jou was ik hier niet geraakt. Bedankt Leo, jouw begeleiding en steun tijdens zowel mijn masterproef als mijn doctoraat waren zeer belangrijk voor mij. Ondanks je drukke schema wist je toch steeds tijd vrij te maken. Bedankt om mij altijd bij te staan met raad en daad. Bedankt Philippe, want je deur stond steeds wagenwijd open. Welke vraag ik je ook toewierp, steeds wist je me terug op weg te helpen. Je hoofd is een groot vat vol wiskundige kennis, en ik ben blij dat ik er soms eens in heb mogen grabbelen. I wish to thank all members of my jury, Aart Blokhuis, Stefaan Caenepeel, Jan De Beule, Maarten De Boeck and Giuseppe Marino, for taking the time to carefully read my thesis and for the comments that helped improve it. Thanks to Michel Lavrauw and Corrado Zanella for showing me the ways of Segre varieties and welcoming me in Vicenza not once but twice. Bedankt Stefaan De Winter om mij uit te nodigen voor een bezoek aan Houghton, om mij te laten kennis maken met coset geometries en blueberry pancakes. Thank you John, for always helping me out, with the big problems and the small, with the mathematical and the non-mathematical. Bedankt aan alle VUB- en UGent-collega’s voor de fijne lunches, koffiepauzes vol roddels, filmavonden,. In het bijzonder wil ik Inneke en Timmy bedanken, wiens verhalen nooit konden vervelen. Bedankt aan alle vrienden en familie voor het vele niet-wiskundige entertainment. Tenslotte, Pieter, jij bent de beste. Bedankt om mij te laten lachen de eerste keer dat we elkaar ontmoette en elke dag sindsdien! Sara Rottey December 2015 v Preface Finite geometry can be described as the study of any particular incidence structure containing only a finite number of points. While there are many structures that could be called finite geometries, attention is mostly paid to substructures of finite projective and affine spaces. Such spaces may be constructed via linear algebra, i.e. starting from a vector space V (n +1, q) of rank n + 1 over a finite field Fq, one can construct an n-dimensional projective space PG(n, q). The affine and projective spaces so constructed are called Desarguesian. At the same time, finite projective spaces can also be defined in a purely axiomatical way. However, while for dimension two there do exist non-Desarguesian planes, for dimension three or greater, any finite projective space arises from a vector space over a finite field. In this branch of finite geometry, different objects of study include vector spaces, incidence structures, affine and projective spaces and various substructures con- tained in them. This area of research started with the seminal work of Beniamino Segre considering algebraic characterisations of combinatorially defined objects. From an algebraic point of view, a conic in a projective plane is an absolutely irre- ducible algebraic curve of degree two. Segre’s celebrated characterisation theorem from 1954 states that in the Desarguesian projective plane PG(2, q), with q odd, every set of q + 1 points, no three of which are collinear (also called an oval), is a conic. However, for q even, not all ovals in PG(2, q) are conics. Even more so, there is still no known classification of ovals in the Desarguesian projective plane of even order q, for q > 4. This problem has inspired the study of other classical sets in projective spaces, such as for example ovoids, i.e. sets of q2 + 1 points in PG(3, q), q > 2, no three of which are collinear. An elliptic quadric Q−(3, q) is a particular example of an ovoid. For q odd, all ovoids in PG(3, q) correspond to elliptic quadrics. However, for q> 32 even, the classification of ovoids remains an open problem. Generalised quadrangles (GQ) are point-line incidence structures that have connec- tions with several other geometrical objects. In particular, ovals and ovoids lead to translation generalised quadrangles. Starting from an oval O in PG(2, q) and an ovoid in PG(3, q), Tits (1959) constructed two GQ’s denoted by T (O) and O 2 T ( ). Payne derivation of T (O), when q is even, yields a third non-isomorphic 3 O 2 GQ written as T2∗(O′), where O′ is the unique hyperoval extending O. A gen- eralisation of this construction is the linear representation T ∗( ) of a point set n K . Several interesting incidence structures, such as semipartial geometries and K (α, β)-geometries, but also interesting graphs, for instance semisymmetric graphs, can be obtained from the linear representation of a well-chosen point set. vii Thas (1974) generalised T (O) and T ( ) by considering the higher dimensional 2 3 O equivalent of (hyper)ovals and ovoids, called pseudo-(hyper)ovals and pseudo- ovoids, both are examples of eggs. One can construct a translation generalised quadrangle T ( ) from every egg ; moreover, every translation generalised quad- E E rangle arises as T ( ) for some egg . The concept of considering a projective E E space over a smaller subfield is called field reduction, this means that points of PG(n 1, qt) correspond to (t 1)-dimensional subspaces of PG(tn 1, q). The − − − set consisting of all these induced subspaces forms a Desarguesian spread. Ap- plying field reduction to ovals and ovoids leads to so called elementary examples of pseudo-ovals and pseudo-ovoids. Of course, eggs obtained in this way do not lead to new generalised quadrangles. Not all eggs are elementary, but a complete classification is not yet attained. A generalised linear representation T ∗ ( ) is an incidence structure similar to a n,t K linear representation, but instead of a set of points, a set of disjoint (t 1)- K − spaces in PG(n, q) is considered. If corresponds to a pseudo-hyperoval, the K corresponding structure is again a generalised quadrangle. If is a (t 1)-spread K − in PG(2t 1, q), this construction corresponds to the Andr´e/Bruck-Bose repre- − sentation of an affine translation plane. When the spread is Desarguesian, the affine plane is Desarguesian. The Andr´e/Bruck-Bose representation thus provides a nice representation of the Desarguesian plane PG(2, qt) as subspaces of PG(2t, q). This representation has proven to be useful for the construction of substructures of PG(2, qt), for instance for the generation of (translation) arcs and unitals, and can contribute to the classification or characterisation of them. This thesis contributes to some of the aforementioned combinatorial questions by focussing on the characterisation of substructures such as pseudo-caps, eggs, spreads, linear representations, subgeometries and unitals. Chapter 1 recalls the basic definitions and fundamental results concerning inci- dence structures and (substructures of) projective spaces over finite fields. Part I considers characterisations of elementary pseudo-caps and Desarguesian spreads. In Chapter 2 we investigate pseudo-caps and (weak) eggs. We pro- vide conditions on element induced spreads which ensure that these structures are contained in a Desarguesian spread. Next, in Chapter 3, focussing on the Desarguesian spread itself, we obtain a geometric characterisation in terms of the normal elements of the spread. In Part II, we consider the linear representation T ∗( ) defined by a point set n K K at infinity. We investigate the isomorphism problem for linear representations in Chapter 4. If the set contains a frame, then the full automorphism group of K viii this structure is obtained. Using the corresponding incidence graph, we construct new infinite families of semisymmetric graphs in Chapter 5. Part III focusses on the Andr´e/Bruck-Bose representation of PG(2, qn) in PG(2n, q). n We investigate the representation of Fqt -sublines and Fqt -subplanes of PG(2, q ) in Chapter 6. In Chapter 7 we obtain a characterisation of the ovoidal Buekenhout- Metz unitals of PG(2, q2) in terms of its Baer secants. Lastly, an extended summary in English and a brief overview in Dutch is given, summarising the main results of this thesis. ix Contents Dankwoord .................................. v Preface .................................... vii Contents.................................... xi 1 Preliminaries 1 1.1 Finitefields ............................... 1 1.2 Incidencestructures .......................... 2 1.3 Projectivespaces ............................ 4 1.4 Field reductionand Desarguesianspreads . .. 14 1.5 The Andr´e/Bruck-Boserepresentation . .. 17 I Pseudo-ovals, eggs and Desarguesian spreads 2 Characterisations of elementary pseudo-caps 21 2.1 Preliminaries .............................. 21 2.2 Desarguesianspreads. 23 2.3 Pseudo-ovalsinevencharacteristic . .. 26 2.4 Characterisationsofgood(weak)eggs . 42 3 A geometric characterisation of Desarguesian spreads 53 3.1 Introduction............................... 53 3.2 Preliminaries .............................. 54 3.3 Spreads of PG(rn 1, q) containing r normalelements .
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