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30.3, q2 ON n-COVERS OF PG(3,q) AND RELATED STRUCTURES by Martin Glen Oxenham, B.Sc.(Hons.) (Adelaide) Thesis submitted for the degree of Doctor of Philosophy at the University of Adelaide Department of Pure Mathematics Novembel 1991 M'illumina d'immenso. Giuseppe Ungarettr, TABLE OF CONTENTS Page INTRODUCTION 1 SIGNED STATEMENT 10 ACKNOWLEDGEMENTS 11 CHAPTER l: PRELIMINARIES 12' 1.1 Finite lncidence Structures 12 1.2 Finite Linear Spaces 14 1.3 Finite Affine and Projective Spaces 16 1.4 k-Caps, Ovaloids and Ovoids in PG(n,q) 21 1.5 Finite Nets and Net Replacement 23 1.6 Hall Triple Systems and Finite Sperner Spaces 25 1.7 Blocking Sets of Finite Affine and Projective Planes 28 1.8 Spreads, Packings and Switching Sets in PG(n,q) 30 1.9 Phlcker Coordinates and the Klein Correspondence 34 1.10 Line Complexes of PG(3,q) 39 1.11 Generalised Quadrangles 43 1.12 Designs 46 CHAPTER ll: n-COVERS OF PG(3,q) 49 2.1 n-Covers of PG(3,q) 49 2.2 General Linear Complexes and Symplectic Spreads of PG(3,q) 52 2.3 Construction Techniques for Proper n-Covers of PG(3,q) 62 Page CHAPTER lll: REGULAR PARTIAL PACKINGS OF PG(3,q) AND PROPER n-COVERS OF PG(3,2) 80 3.1 The Uniqueness of the Proper 2-Cover of PG(3,2) 80 3.2 On a Representation of the Proper 2-Cover of PG(3,q) and a Classical Group lsomorphism 91 3.3 Regular Partial Packings of PG(3,q) and Blocking Sets of PG(2,q2) 98 3.4 On the Classification of the Proper n-Covers of PG(3,2), n ) 3 108 CHAPTER lV: QUASI-n-MULTIPLE DESIGNS AND n-COVERS OF PG(3,q) 124 4.1 Quasi-n-Multiple SpernerDesigns 124 4.2 Characterising Finite Affine Spaces and Embeddings of Quasi-Multiple Affine Designs 132 4.3 Hall Triple Systems and the Burnside Problem 139 4.4 Resolution Classes of Balanced lncomplete Block Designs and Partitions of Finite Affine Spaces 152 CHAPTER V: COMPATIBLE AFFINE PLANES AND NET REPLACEMENT 161 5.1 On the Relationship between the Embedding of a Finite Affine Plane in a Second Affine Plane of the Same Order and Net Replacement 161 5.2 Redei Blocking Sets, Replacement Nets and Switching Sets 169 5.3 On the Compatability of Finite Affine Planes 173 5.4 On Blocking Sets in Affine and Projective Planes 178 CHAPTER Vl: CONCLUSION 182 BIBLIOGRAPHY 185 INTRODUCTION In the realm of finite projective geometry, the notion of a spread has proven to be of fundamental importance and has been the driving force behind a sizeable amount of the research carried out in this field, especially during the second half of this century. Much of the interest in spreads has arisen from the pioneering work of André in [2] and also of Bruck and Bose in [19] and [20] who dernonstrated the usefulness of spreads in the construction of finite translation planes. The simplest examples of spreads a,re those of the finite three dimensional projective space PG(3,q). In this case a spread consists of. q2 * l pairwise skew lines. The principal topic of this thesis concerns generalisations of the spreads of PG(3, q); these generalisations, referred to by Ebert in [SZ] and [53] as r¿-covers of PG(3, g), are sets of lines which satisfy the property that each point of PG(3, q) is incident with exactly n of these lines. Sets of lines of PG(3,q) satisfying this condition have also been referred to as n-fold spreads of 1-spaces (see [71]). We note before proceeding, that in [14], Beutelspacher also speaks of n-covers of PG(ú, q); his definition, however, is used in quite a different context and so is not relevant to what we discuss here. Ebert's motivation for studying ??-covers of PG(3, q) (see [53]) arose from consider- ing the problem of completing partial packings of PG(3rg) to packings. Given a partial packing of PG(3, q), the complerrent of it in PG(3, q) is an r¿-cover for some integer rz. The problem of completing the partial packing then becomes one of partitioning the r¿-cover into n spreads. Ebert therefore clefined a proper rz-cover of PG(3, g) to be one which cannot be so partitioned. Our motivation is slightly different. We are more interested in studying n-covers for their own sake, that is to say our main aim in this thesis is to develop a theory of. n- covers which parallels that of the spreads of PG(3, q). A. a part of this theory involves 1 examining the designs which result from the r¿-covers, we have found it necessary to alter the definition of a proper n-cover. In this thesis, a proper r¿-cover of. PG(3,q) is one which is not the disjoint union of two ni-covers with n1 t nz : n. The reason for this alteration will become evident when we consider the decomposability of the aforementioned designs. Regarding the existence of n-covers of PG(3, g), they exist for all admissible values of n, that is for all n from ! to q2 -f q * 1. This is an immediate consequence of the existence of packings in PG(3, g), (see, for exampte [13] by Beutelspacher and [46] and [a7] by Denniston), as any subset of n spreads in a packing constitutes an ??-cover. The first infinite class of proper ??-covers was discovered by Ebert and appears in [53]; the ??-covers in this class are all 2-covers and exist for every odd prime power g. His construction relies on a partition of the point-set of PG(s,q) bV ovoids which he established in [52]. The 2-covers then result by exploiting the manner in which certain lines of PG(3,g) intersect the ovoids. He also provided examples of proper 2-covers for g:2,4 and 8. Since these are constructed by means similar to those used in the case that q is odd, there is good reason for believing that they too lie in an infinite class, but this is yet to be resolved. The chief goal of Chapter II of this thesis is to describe and explore possible alter- native techniques for constructing proper ??-covers of PG(3,q). These mainly involve beginning with a known (q * l)-cover of PG(3, q), from which rvr/e remove maximal sets of pairwise disjoint spreads. This either yields a partition of the (q f 1)-cover into spreads or it produces at least one proper ??-cover with 2 1n 1 q+L.The three types of (q + l)-covers we consider are the set of lines of a general linear complex, the set of lines in the quadratic complex consisting of the tangent lines to an elliptic quadric and the set of lines in a long Singer orbit. Using these sets, we obtain explicit examples of proper z¿-covers for some small values of q. Our main result, based on a theorem of 2 Thas on generalised quadrangles (see Theorem 3.4.1 of [10a]), is that proper rz-covers exist in PG(3,q), g even for some n satisfying2 < n 1q. In studying r¿-covers of PG(3, q) the major obstacle to overcome is that of actually determining whether or not a given ??-cover is proper. A very basic sufficient condition for an n-cover not to be proper is that it contain a spread, this condition also being necessary for n equal to 2 and 3. The question of the existence of spreads in the quadratic complex (described in the previous paragraph) and in a long Singer orbit, is indirectly addressed in part by the work of Ebert in [53] and [54]. For general linear complexes, much more is known. The only existing examples of spleacls lying in a general linear complex (also called symplectic spreads) are the follor,ving: The regular spreads for all values of g (see, for example, [51]), tlìe Lüneburg spreads for q - 22hrt , h > I (see [90] and [91]) and finally, for g odd and non-prime, a class of spreads constructed by Kantor in [82] which gives rise to a family of Knuth semifield planes. As a supplementary fact, we prove that a subregular spread which is not also regular, is not symplectic for any value of g. In [61], Glynn showed that trvo regular spreads lying in a general linear complex of PG(3, g), q even intersect in either a single line or in the q + 7 lines of a regulus. By using a different argument (which is independent of the parity of q),we prove that the result also holds for g odd. For the case that g is even, stronger results have been obtained by Bagchi and Sastry in [3]. In particular, by dealing with an equivalent problem, they implicitly proved that any spread lying in a general linear complex Lo of PG(3,g), g even meets every regular spread and every Ltineburg spread lying in Lo, in at least one line. They have also implicitly classified the intersection patterns of the regular and Lúneburg spreads lying in a comrnon general linear complex (see [ ]). By applying these techniques and results, we construct four explicit examples of proper ??-covers. The first two are a proper 2-cover and a proper 3-cover of PG(3r2).
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