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## Material Information- Title:
- Generalized quadrangles of order (s, t) with [s-t]=2
- Creator:
- Miller, Mark Anderson
- Publication Date:
- 1999
- Language:
- English
- Physical Description:
- ix, 150 leaves : ; 28 cm
## Subjects- Subjects / Keywords:
- Finite generalized quadrangles ( lcsh )
Finite generalized quadrangles ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 146-150).
- General Note:
- Department of Mathematical and Statistical Sciences
- Statement of Responsibility:
- by Mark Anderson Miller.
## Record Information- Source Institution:
- |University of Colorado Denver
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- Auraria Library
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- Resource Identifier:
- 44077697 ( OCLC )
ocm44077697 - Classification:
- LD1190.L622 1999d .M55 ( lcc )
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GENERALIZED QUADRANGLES OF ORDER (S, T) WITH \S-T\ = 2 by Mark Anderson Miller B.S.E., John Brown University, 1988 M.S., University of Colorado at Denver, 1993 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics 1999 This thesis for the Doctor of Philosophy degree by Mark Anderson Miller has been approved by Stanley E. Payne William E. Cherowitzo Sylvia A. Hobart J. Richard Lundgren William J. Wolfe Date Miller, Mark Anderson (Ph.D., Applied Mathematics) Generalized Quadrangles of Order (s,t) with \s t\ =2 Thesis directed by Professor Stanley E. Payne ABSTRACT The subject of this thesis is generalized quadrangles (GQ) whose pa- rameters differ by two. The roles of regular points, lines and ovoids are ex- amined extensively. Such regularities allow for connections between various GQ. Affine planes associated with GQ(q + 1 ,q 1) are shown to be iso- morphic to those associated with GQ(q,q). Using a new idea, the grid-like axiom, the affine planes associated with certain regular ovoids are shown to be isomorphic. While regular ovoids can be used to obtain numerous other ovoids, the fan containing a pivotal ovoid is shown to be unique. Moreover if a fan contains more than one regular ovoid it is shown to contain every regular ovoid of the GQ. A new characterization of the GQ(q + 1, q 1) arising from a g-arc is provided. After considering the characterizations due to De Soete and Thas and to Payne, the standard coordinatization of the GQ is introduced. By examining the coordinatizing held permutations, Paynes sixth characterization axiom is shown to be equivalent to having desarguesian planes arise from the pivotal ovoids. Collineations of the known GQ(q + 1, new result here is that in only one case are there collineations which are not induced by semilinear maps of the underlying vector space. An appendix is provided as a brief introduction to projective geometry with a list of the known hyperovals. This abstract accurately represents the content of the candidates thesis. I recommend its publication. Signed Stanley E. Payne DEDICATION This thesis is dedicated to my mother Janet, my sister Michelle, and my wife Adriana. ACKNOWLEDGMENTS An endeavor of this magnitude of course leaves many people to thank. I would like to acknowledge my thesis committee and thank them for their willingness to undertake this project with me. In particular I thank my advisor and mentor Stanley Payne for his tireless efforts to aid me in becoming a mathematician, and I thank Bill Cherowitzo who has acted as my co-mentor here at CL-Denver. While in progress, portions of this thesis were presented at the Dis- crete Mathematics Seminar at the University of Colorado at Denver, the Al- gebraic Combinatorics Seminar at Colorado State University, and an AMS Special Session in Groups and Geometry at Kansas State University. I would like to thank the various organizers, Richard Lundgren, Robert Liebler, and Ernest Shult for giving me the opportunity to present and refine this material. My appreciation is extended to the CU-Denver Mathematics Gradu- ate Committee for the financial assistance I received here which allowed me to pursue my research. Moreover, I gratefully acknowledge the generosity of the Warren Bateman Family exhibited through their funding of the Bateman Teach- ing Fellowship of which I was a recipient. This fellowship was established in memory of Lynn Bateman whose reputation as an outstanding mathematics educator inspires us all. Over the past four years many friends and family members have pro- duced an immeasurable amount of encouragement. While I cannot name them all here, I do wish to thank a few specific individuals whose tangible support proved to be both necessary and sufficient: C.B. and Tonny Euser, Allen and Leanne Holder, Matthew and Kimberly Lockhart, Michael and Nancy Miller, and my parents, Roy and Janet Miller. Finally I thank my wife Adriana whose constant, unfailing love has helped in ways I cannot express. I am forever in your debt. M. A. Miller Denver, Colorado CONTENTS Chapter 1. Introduction and review ........................................... 1 1.1 Preface............................................................... 1 1.2 Historical background ................................................ 4 1.3 Definitions, examples, and observations............................... 5 2. Known examples ................................................... 16 2.1 Two classical examples: W(q) and Q(4, q)............................. 16 2.2 A construction by Tits............................................... 17 2.3 Constructions by Ahrens and Szekeres ................................ 20 2.4 A construction by Payne.............................................. 23 3. Regular ovoids in S............................................... 29 3.1 Affine planes and ovoids............................................. 29 3.2 Intersection of traces with ovoids of M.............................. 32 3.3 Constricting about a regular ovoid .................................. 34 3.4 Regularity in Soo and planar isomorphisms............................ 35 3.5 Grid-like Fans....................................................... 41 viii 4. Regularity in S..................................................... 50 4.1 A theorem of Thas and van Maldeghem .............................. 50 4.2 Regular pairs..................................................... 60 5. Characterizations................................................... 69 5.1 (0,2)-sets........................................................ 69 5.2 Axioms of Payne................................................... 91 5.3 Coordinatizing the GQ............................................ 104 5.4 Relating Paynes Axioms to a and /?.............................. Ill 6. Collineations...................................................... 118 6.1 Collineations of 7. Other constructions................................................ 126 7.1 Constructions using groups and cosets ........................ 126 7.2 Constructions using groups and designs........................ 132 Appendix A. Projective planes and 3-space..................................... 136 B. Arcs and ovals.................................................... 141 References............................................................ 146 IX 1. Introduction and review 1.1 Preface While countless articles have been published on the topic of general- ized quadrangles, relatively few self-contained references in this area are avail- able. One notable exception is Finite generalized quadrangles by Payne and Thas which was published in 1984. Since then there has been somewhat of an explosion of interest in GQ. It now appears that due to the great proliferation of work in this held no suitable sequel would be fathomable. So it seems that a better task might be collecting much of what is known about a particular type of GQ and providing a sort of specialized reference manual. This is the motivation for what follows with the focus on generalized quadrangles whose parameters differ by 2. The purpose of this thesis is to gather together in one place much of the relevant work to date involving these GQ, including work which originated with this author. Portions of the material here were previously reported with Payne in [PM98]. Chapter one provides an introduction to generalized quadrangles with 1 some basic definitions and small examples. In chapter two the known exam- ples are presented as are properties of spreads/ovoids in these examples. In particular the hyperbolic lines in the pivotal ovoids are seen to be related to affine lines and planar arcs. Chapter three deals primarily with regular ovoids of GQ(q + 1, q 1) and the corresponding affine planes. In addition to the known result that pivotal ovoids correspond to regular points, the following material is new: (1) The planes corresponding to ovoids are shown to be isomorphic, and sometimes identical, to planes associated with GQ of order q in theo- rems 3.4.4 and 3.4.5. (2) The Grid-like Axiom is introduced in section 3.5. (3) Every regular ovoid is shown to be pivotal for exactly one fan in theo- rem 3.5.4. (4) Planes associated with ovoids of a grid-like fan are shown to be iso- morphic in theorem 3.5.5. Chapter four begins with an elaboration of the proof by Thas and van Maldeghem showing that no non-classical GQ(q + 1 ,q 1) can have all points regular. Next regular pairs of points are examined leading to a new classification of regular ovoids. In chapter five, a new characterization of the GQ(q + 1, q 1) arising 2 from a q-arc is provided. After considering the characterizations due to De Soete and Thas and to Payne, the standard coordinatization of the GQ is introduced. By examining the coordinating held permutations, Paynes sixth characterization axiom is shown to be equivalent to having desarguesian planes arise from the pivotal ovoids. Collineations of the known GQ(q+1, q1) are discussed in chapter six. The main new result here is that in only one case are there collineations of the quadrangle which are not collineations of projective three-space. The chapter concludes with a clarification of remarks made in [PM98] regarding findings of [GJS94], In Chapter Seven, two previously known alternative constructions are provided: one due to Payne and the other to De Bruyn. An appendix is provided as a brief introduction to projective geometry with a list of the known hyperovals. While no work can be completely self-contained, an attempt has been made to keep the reader from having to consult an unwieldy number of refer- ences to understand the material here. Most terms not defined in this thesis can be found in either [Dem68] or [PT84], A nice overview of generalized quad- rangles is given in [Tha95] as well as [Pay96], the former of which concerns generalized n-gons with n not necessarily equal to 4. 3 1.2 Historical background The study of finite generalized quadrangles (GQ) is a relatively young branch of discrete mathematics. J. Tits first introduced the notion of a gener- alized polygon in 1959 [Tit59]. For the next decade some progress was made in this field, particularly in the study of what are now termed the classical gener- alized quadrangles (see for example [Dem68] and [FH64]); then from the late 1960s through the early 1980s many geometers and algebraists began looking more deeply at this subject. A plethora of new results and a good number of new examples were discovered. As the focus of this thesis is on GQ(s,t) where s and t differ by two, some background on these GQ may be in order. (The roles of s and t are described in the next section.) In the late 1960s and early 1970s new constructions for GQ(s,s + 2) were given by Ahrens and Szekeres in [AS69] and independently by Hall in [Hal71]. This inspired Paynes work in [Pay71b] which gave new GQ(s, s 2). With these constructions in hand, two questions naturally arise: What are the collineations of these GQ, and what are the defining characteristics of these GQ? Much work has been done in an attempt to answer these questions. These two questions serve as a motivation for much of this thesis. 4 1.3 Definitions, examples, and observations We begin with some basic definitions and then look at a few small examples. Let V and B be two non-empty sets, called points and lines, with an incidence relation X such that there are positive integers s and t satisfying Gl) Each point is incident with f+1 lines; any two points are mutually incident with at most one line. G2) Each line is incident with s +1 points; any two lines are mutually incident with at most one point. G3) Given a line L and a point x not incident with L there is a unique point y and a unique line M such that x X M X y X L. Such a collection S = (V, B.X) is called a generalized quadrangle of order (s,t) written GQ(s,t); when s = t the GQ is said to have order s. The dual of a GQ(s,t) is the GQ(t,s) obtained by interchanging the roles of points and lines. Any theorem or definition given for a GQ can be dualized by interchanging points and lines. It will be assumed that whenever a definition or theorem is given, its dual has also been given. Two points incident with a common line are collinear and two lines incident with a common point are concurrent, x ~ y means that x and y are either collinear if x and y are points or concurrent if x and y are lines. If X is a set of points (respectively, lines) of S, then X1- denotes the 5 set of all points collinear (resp., lines concurrent) with everything in A"; X1- is called the trace of X. If X = {a;} is a singleton set, it is common to write X1- as xL. The span of X. written Xx, is the set of all points collinear (resp., lines concurrent) with all of X. By convention, x x. It is worth noting that (X^)1- = XL. Given any two noncollinear points x and y, {agt/}-11- is called the hyperbolic line through x and //. Dually the hyperbolic point on two nonconcurrent lines L and M is the span {/,. .1/} . Often GQ under examination here will be related to projective geome- tries. To avoid confusion, the symbol < x, y > will be used for the projective space spanned by x and y whereas xy will indicate the line of the GQ con- taining x and y. Sometimes the term Straightforward arguments demonstrate the following: For x t- 'P. |re-11 = st + s + 1. For L E B, = st + t + 1. \P\ = (s + l)(st + 1). \B\ = (t + l)(st + 1). For two noncollinear points x, y, |{rc, t/}| = t + 1 and 2 < |{rc, t/}J"L| For two nonconcurrent lines L, M, \{L, M}\ = s + 1 and 2 < \{L, M}J-\ < s + 1. Two noncollinear points form a regular pair provided their span attains the upper bound, t + 1; two collinear points are dehned to form a regular pair. An individual point is a regular point provided it forms a regular pair with every other point. Dually, { /.. M} is a regular pair of lines provided either { L. M}J- = s + 1 or L and M are concurrent; a regular line is one which forms a regular pair with every other line. A point is coregular provided all lines incident with it are regular. The following three observations will be useful when working with regularity. Observation 1.3.1 {o^o^} is regular if and only if whenever {2/1,2/2} ^ Ui-x-,} then {yi,y2} = {xi.x2} Proof: If Xi ~ x2, then clearly the observation holds as {xi,x2} = {.r 1. Now assume x\ and x2 are not collinear. Suppose {a; 1,0:2} is regular and let {.rt} = {0:1,.... xt. 1}. If {2/1,2/2} Q { c 1. } then each Xi is in {yi,y2}L. Because \{yi,y2}L\ = t + 1, {2/1, y2}L = {xi, x2}L. Now suppose {2/1,2/2}^ = { c 1. } whenever {2/1,2/2} Q This forces |{a;i,a;2}| = t + 1 in which case {x\,x2} is regular. 7 Observation 1.3.2 If {xi,x2} is regular then zi,z2 E {.r if and only if {zi,z2} = {./i. } . Proof: The proof is similar to that of observation 1.3.1 and is left to the reader. Observation 1.3.3 A point X\ is regular if and only if every pair of points in is regular. Proof: Assume x\ is regular. Let yi,y2 E xf, x2 E {yi, By Observation 1.3.1, {yi,y2)'L = Hence Therefore {yi,y2} is regular. Conversely, assume every pair of points in Xi is regular. Let x2 be some point other than x\ and let {yi,y2} C {xi^x^^. By Observation 1.3.1, as {x\,x2} C {yi,y2}, is regular. Therefore x\ is regular. In the future, the proofs of numbered observations will be left to the reader. An ovoid is a collection of st + 1 pairwise noncollinear points. For k < st + 1, a k-cap is a set of k pairwise non-collinear points. A spread is a collection of st + 1 pairwise nonconcurrent lines. A set of ovoids which partitions the point-set is a fan. A set of spreads 8 which partitions the line-set is called a packing. An ovoid (resp. spread) is said to be regular if the span of any pair of its points (resp. lines) is of maximum size and is contained in the ovoid (resp. spread). A GQ(s,t) with s or t equal to 1 is called thin; otherwise the GQ is thick. A smallest thick GQ would have parameters s = t = 2. Such a GQ is constructed below. For the most part, the GQ examined in this thesis will be thick. However, it is sometimes instructive to begin with small, thin examples. Example 1.3.4 The 4x4 Grid: Let B be the lines of a 4 x 4 grid and let V be the points of intersection on the grid. Observe that this gives a GQ(3,1). The next example is the dual of the first. Example 1.3.5 I\ u (The Complete Balanced Bipartite Graph on 8 Nodes): Let V be the node set of /\ u and let B be the edge set. This gives a GQ( 1,3). In general for n > 1, any n x n grid is a GQ(n 1,1) whose dual is Kn,n (sometimes called a dual grid) which is a GQ(l,n 1). However the case n = 4 is of particular interest here. To see why, first consider the smallest thick GQ. Example 1.3.6 GQ(2,2): Let S = {1,2,..., 6} and let V = { {i,j}\i,j E S,i ^ j} be the set of duads, i.e. subsets of size 2, from S A syntheme is a triple of disjoint members 9 of V. For example {{1, 2}, {3, 4}, {5, 6}} is a syntheme. Let B be the set of all synthemes formed from the duads in V. Straightforward verification shows that S = (V, B, e) is a GQ of order 2. Sometimes it is helpful to view a particularly GQ from more than one point of view. The GQ(2, 2) may also be viewed differently using algebra. Let Z2 be the group of order 2 and set G = Z2 x Z2 x Z2. For i e {1, 2, 3} let e* be the element with a 1 in the ith position and zeros elsewhere, let e be the identity element, and let j = J2ei- Construct the following subgroups of G: A* = {e, e*}, A* = {e, and the set T. Let the lines of S be the subgroups in F together with their cosets. If incidence is given by containment and inclusion, then S is isomorphic to example 1.3.6. Similarly, examples 1.3.4 and 1.3.5 may be viewed from an algebraic point of view. Before this is done a relationship between the three examples is established. At first glance examples 1.3.4 and 1.3.5 may seem unrelated to 1.3.6. However, there is a very elegant connection. Let S be a GQ(2,2) with a point x, and let Px be the points of S which are not in x. Let Bi be the lines of S which do not contain x (note that each of these lines has a point 10 removed from it). For each y E Px, let Ly be a new line which joins y to the unique remaining point of {aqi/jT-1, and let B2 be the set of all such Ly. Let Bx = Bi U B2. Then Px and Bx form the point and line sets (respectively) of a GQ( 1,3). Now return to S and let L be any line of S. We create a similarly derived structure as follows. Let BL be the lines of S which are not in IT; let Pi be the points of S which are not on L. For each M E BL, let xm be a new point incident with the unique remaining line of {L. M} Define P2 to be the set of all such with PL = Pi U P2. The resulting structure with point set Pl and line set BL is a GQ(3,1). This relationship between the three GQ is not coincidental; in fact it lies at the heart of this thesis. For this reason this process is explained in more detail here. Let S = (V,B,X) be any GQ(q,q) with a regular point x. Let Vx = V xL. Let Bx = Blx U Bj. where Blx is the set of lines which are not incident with x, and B~ = { {x/yj^ly E V x }. i.e. B2X is the set of hyperbolic lines through x. If the incidence Xx is given by that of X and by containment, it is easy to verify that the resulting structure Sx = (Vx, Bx,Xx) has q points on each line, and each point of Sx is on q + 2 points. That G3 holds is shown below in cases. (1) Let L eB\ and let p be a point of Vx not on L. In this case p and L 11 are also elements of S. There is a unique line M E B incident with p and a point z of L. (a) If z E Vx, then M is the unique line of B\ collinear in Sx with p and a point of L. If some hyperbolic line of B2 confined p and a point tv of L, then tv and p would both be collinear in S with a point of xx which would give a triangle in S. Hence M is the unique line of Bx through p and a point of L. (b) If z 0 Vx, then z E {x,p}. As x is regular, every line through z contains a unique point of {.r.p) Specifically, L contains a unique point tv of {.r.p) The hyperbolic line {.r.p} is the unique line of Bx containing p and a point of L. (2) Let L E B2 and let p be a point of Vx not on L. Counting the number of points in Sx gives \PX\ = \P\ = (q + l)(q2 +1) (q + l)q 1 = q3. In S there are q points in L\{a;}. Each of these points is on q +1 lines. As no point of Vx can be on more than one such line, these lines cover q(q + l)(q 1) = q3 points of VX\L\ i.e. these lines partition the points of VX\L. Hence there is a unique line M E Blx which is incident with p and a point of L. Because p 0 L, the regularity of x implies {.r.p) has no point of Vx in common with L. Therefore M is the unique line of Bx which is 12 incident with p and a point of L. This shows that Sx is a GQ(q 1 ,q + 1). Most GQ constructions presented here are similarly straightforward to verify. For this reason it will be often left to the reader to check that a given incidence structure satisfies Gl, G2, and G3. The process of forming Sx from S is called expanding about the regular point x. Sx is sometimes written P(S,x). Dually, if L is a regular line of S, replacing x with L and interchanging the roles of points and lines gives a GQ(q + 1, P(S, L). Naturally this process is called expanding about the regular line L. If these ideas are interpreted algebraicly, observe that the 4x4 dual grid can be constructed using groups. Let P+ = T U {At} where A = {g j}- The points of the dual grid can be viewed as points of G\ the lines can be viewed as cosets of subgroups in P+. The 4x4 grid can be constructed similarly. Let T' = {Mi, A, AA, A A}. The points of the grid are the elements of G and the cosets of subgroups in A. Section 7.1 of this thesis explores this interpretation further. In some respects expansion about regular points and lines is the GQ- analogue to the relationship between projective and affine planes. If 7r is a 13 projective plane, a line L and all of the points on L can be removed from n to form the corresponding affine plane. Similarly, a point p and all of the lines through p can be removed from tt to form the corresponding dual affine plane. We know that to every projective plane of order n there corresponds an affine plane of order n. Every known GQ(q, q) has at least one regular point or one regular line, and hence gives constructions of GQ(q 1,^ + 1) or GQ(q + 1 ,q 1). Moreover, every known GQ whose parameters differ by 2 arise by the method of expansion. However, expanding about two different regular points may produce to two non-isomorphic GQ. Recall that every known projective plane has prime power order. Sim- ilarly in every known example of GQ with parameters (s, s), (s 1, s + 1), or (s + 1, s 1), s is a power of a prime. This may seem unusual at first reading, however further examination shows a strong relationship between projective geometries and GQ. We close this chapter with some special constructions of planes relat- ing to GQ. Readers unfamiliar with projective and affine planes are directed to the appendix. Let S = (V, Â£>,Z) be a GQ(q, q) with a regular point p incident with a regular line L. Construction 1.3.7 The projective plane ir*(p) = ( V*(p),C*(p) ): Let T*{p) = p. Let C*(p) = {{x,y}\x,y e ir1}, i.e. C*(p) consists of the 14 contains q + 1 points. Because any pair of points in V*(p) determines a unique member of Â£*(p), ir*(p) is a projective plane of order q. Construction 1.3.8 The affine plane ir(p) = ( T(p),C(p) ): Let C(p) = Â£*(p)\{L}, and let T{p) be the points of T{p) with the points of L removed. This makes 7r(p) an affine plane of order q. Construction 1.3.9 The projective plane it*(L) = ( T*(L),C*(L) ): Let V*{L) = { { M. iV}-L|M, N e L } and Â£1 (I.) = L. Incidence is given by containment. This is a projective plane of order q. Construction 1.3.10 The affine plane 7r*{L) = ( Â£>*(Â£),Â£*(Â£) ): Remove L and all of its points to form the affine plane 7r* (L) = ( V* (L), Â£* (L) ). The projective planes constructed in 1.3.7 and 1.3.9 are well known. The related affine planes of constructions 1.3.8 and 1.3.10 will be revisited in section 3.4. 15 2. Known examples In this chapter a number of GQ will be constructed from objects in projective 3-space. For these constructions some notation for the projective objects is first established. See the appendix for more on projective planes, projective 3-space, ovals, and oval permutations. Let F = GF(q) and let a be an oval permutation of F. Let A = (0,1, 0, 0),B = (0, 0,1, 0), Q = {ws = (l,s,s,0)|s e F} U {A}; thus fi is an oval in the plane 11^ = [0,0,0,1]T embedded in PG(3,q) = Q. Let fi+ = fi U {!?} and let fi- = If q is even then B is the nucleus of fi. 2.1 Two classical examples: W(q) and Q(4, q) For two classical examples we visit polar spaces. Let v be a symplectic polarity of Q. Hence for a point .r f Q. .r1' is a plane containing x. For example let v be given by the alternating form g{u,v) = uMvT where M is the skew- symmetric matrix: 0 0 0 1 0 0 10 The lines of Q which are fixed by v 0 -1 0 0 1 0 0 0 are called totally isotropic lines. The points of Q together with the totally 16 isotropic lines of Q under v form a GQ(q, q) written W(q) in which every point is regular. (The polarity u is suppressed in the notation because all such GQ are isomorphic.) Now let f : F5 ^ F be some non-degenerate quadratic form and let Q be the associated quadric. This means b(x,y) = f(x + y) f(x) f(y) is a symmetric bilinear form, and () = {.r e- PG(A, q)\f(x) = 0}. That / is non-degenerate means that for each x there is at least one y such that f(x + y) ^ f(x) + f(y). The points of Q together with the projective lines whose points are all in Q also form a GQ(q,q), written Q(4, q). The two GQ, W(q) and Q(4, q), are point-line duals of one another. They are self-dual exactly when q is even [PT84], 2.2 A construction by Tits The first non-classical example we will examine is one due to Tits. It was first reported in [Dem68]. In [Pay85a] Payne modified Tits description slightly. It is this modified description which is presented here. Define an incidence structure S = where points are of the following two types: i) Points of and ii) Planes of Q which contain either exactly 1 or q + 1 points of fi. 17 The lines of S are the lines of Q which meet 0 in exactly one point. Incidence is given by that of Q. Let wq, ... ,wq be the points of fi with the respective tangent lines labeled LQ,... ,Lq. Verification that Gl, G2, and G3 hold follows from well known counts from projective geometry. These are outlined below. If p is a type i) point of V, then the lines of B through p are exactly the q + 1 projective lines < p,wk >,0 < k < q + 1. If p is the plane 11 v. then the lines of B incident with p are the q + 1 tangents to fi. Finally if p is any other type ii) point, then in Q, p contains exactly one point of fi, say Wj. The lines of B incident with p are the q + 1 projective lines through wj in the plane p. If L = Lk is a tangent to fi in 11 v. then the points of V incident with IIoo are the q + l projective planes through Lk. If L is a line which meets in the point Wj, then the type i) points incident with L are the q projective points of L\{wj}, and the unique type ii) point incident with L is the plane through L and Lj. Let L be a line of B, let pi be a type i) point not incident with L, and let P2 be a type ii) point not incident with L. Projectively, let L meet fi at wj, and let M =< p\,Wj >. As M and L meet in S at the type ii) point < L, M >, M is the unique line of B incident with pi and a point of L. If L is 18 a tangent line to fi, let M be the line of p2 which is also tangent to fi; in this case L and M are both incident in S with the point Iloo. If L is not a tangent line then in Q, L meets Iloo at a point ir r If p2 = 11 v then let M = Lj: otherwise if p2 fi fi = Lk, let y be the projective point common to L and p2 and let M =< y,wk > In either case M is the unique line of B incident with p2 and a point of L. With Gl, G2, and G3 satisfied, this construction is shown to yield a GQ(q, q) which is often written T2(fi) or T(tt) is different from W(q) and Q(4, q). Assume for the remainder of this section that q is even. In this case B is the nucleus of fi. Proposition 2.2.1 Iloo is a regular point of T2(Q). Proof: Let x be a point of T2(Q) not < x,B >, and {11 v../ } is the set of q projective points on < x, B > other than B together with Iloo. Thus {11 v../ } is as large as possible. Proposition 2.2.2 The lines of Iloo which are tangent to fi are regular lines of r2(n). Proof: Let L a line of Iloo tangent to fi at p and let M be a line 19 not coincident in T2(fi) with L; this means M is a projective line meeting 0 at some point r ^ p. Observe that {L, .1/} is the set of q lines spanned by p and a point of M other than r together with the line < B,r >; and {L, M}J- is the q lines through r in < M,p > other than < r,p > together with L. 2.3 Constructions by Ahrens and Szekeres Ahrens and Szekeres provided two examples of non-classical GQ which we describe here. For q odd the original description of AS(q) is as follows: Points of AS(q) are the points of affine three-space AG(3, q). Lines of AS(q) are the following curves of AG(3, q): (i) x = a, y = a, z = b, (ii) x = a, y = a, z = b, (iii) x = ca2 ba + a, y = ^2ca + b, z = a. Payne has given the following alternative construction of this GQ(q 1,5 + 1): Choose any point x of W(q) (recall x is regular in W(q)). Let Vx be the points of W(q) which are not contained in a totally isotropic line through x. Let Â£>,. = B\ U Â£>2 where B\ is the set of totally isotropic lines of W(q) and Â£>2 = {{+ y}x\y E Vx}. Incidence is given by containment. Notice that this is the process of expanding W(q) about x which gives the GQ(q 1,5 + 1) 20 written P(W(q),x). This construction works regardless of the parity of q. For the remainder of this thesis the case where q is even will be of particular interest. For q even this next construction also yields a GQ(q 1,5 + 1). It is a variation on the construction of T2(0) due to Ahrens and Szekeres [AS69] and independently to Hall [Hal71]. Let S+ = (V+.JB+.X+) be the incidence structure in which V+ consists of the type i) points of V, and B is the set of lines of Q which meet in a unique point of fi+. It follows that S+ is a GQ(q 1,5 + 1) using the incidence of Q. Such a construction is written T*(tt+) or sometimes simply S(Q+). Notice that T2(Q+) = P(T2(Q), IIoo). The following observations are immediate. Observation 2.3.1 For each x E fi+, the lines of through x form a spread. Observation 2.3.2 The set of all such spreads formed above is a packing M. Further examination reveals Proposition 2.3.3 If Li,L2 are in a spread of M, then {Li, L2} is regular. Proof: Let Lx fi L2 = y. Li, L2 determine a plane n meeting in a line L through y. This line meets some other point x ^ y of fi+. {Li,L2} is then the set of q lines of fI\{L} through x. Hence {Li, L2}LL is the set of q lines in L1\{L} through y. 21 This packing in fact has a stronger property than just having regular pairs within spreads. Observe that every pair of lines from a spread S of M has its trace contained in some other spread of M and has its span back in S. Such a spread S is said to be pivotal for the packing M. Proposition 2.3.4 If Li,L2 are in different spreads of M. then {Li,L2} = {LuL2} Proof: Assume Sj is a spread of M. containing L\ and assume S2 is a different spread of M containing L2. Let L e {Li, L2}\{Li, L2}. First sup- pose {Li,^}1- contained two lines Mi,M2 from some spread S in M. Then as S is pivotal Li,L2 are contained in a spread of Ad, a contradiction. So has one member in each spread of Ad\{Sj, A2}. As spreads contain only non-concurrent lines, L is in either Sj or S2. But as {L\,L2} is regular, { /.i. } = { /.i. /.} = { /.} . Hence {Li,L2} is contained in either Si or S2, a contradiction. Observe that nothing in the proof above relied on the specific con- struction of the quadrangle, rather it relied only on having a GQ with a packing of pivotal spreads. 22 2.4 A construction by Payne This section provides a construction of GQ(q +1, q 1) for q even due to Payne [Pay71b, Pay72b, Pay85a] which also uses the hyperoval fi+. When fi+ does not contain a conic, the GQ constructed here are actually different from the duals of those constructed by Ahrens and Szekeres. Let Oab (respectively 0Ba) be the set of planes in Q which meet A but not B (respectively B but not A). Finally let = T+UOab^Oba and B be the set of lines in which contain a point of Q_. Again incidence is that of Q. This GQ is often denoted Any two planes of Oab meet in a projective line. Such a line cannot be a line of If L is a line of PG(3, q) the set of planes through L is called the dual line in PG(3,q). If one of these planes is removed, the result is called a dual line in AG(3,q). For any two points x0,y0 E 11 x.. let Vxo,yo be the dual line in AG(3, q) consisting of the q planes other than IIqq which contain the line < xq, t/o > In particular observe that Tab = {lb = [1, 0, 0,i]T\i E F}. Observe that no projective line of Lb is a line of of Each fl* has q2 + q + 1 projective points, q2 of these are pairwise non- collinear points of the quadrangle, and the remaining q + 1 projective points are not part of the GQ. Furthermore each fl* has q2 + q +1 projective lines and no quadrangle lines. Consequently, the fl* will be viewed both as projective and quadrangle objects, where the context will dictate meaning. The dual notion of a pivotal spread is a pivotal ovoid; i.e. an ovoid O is pivotal for a fan M. provided every pair of points in O is regular, every pair in O has its trace in some other ovoid of M. and every pair in O has its span in O. This leads to the next proposition. Proposition 2.4.2 The ovoids Oab,0Ba are pivotal for A4. Moreover the trace and span of any pair within one of these ovoids are affine lines or dual lines in AG(3, q). Proof: Choose ws to be a projective point of Or, and let Pi,P2 Â£ Va,ws- As points of of S, Q and Pi are collinear by a line of S. Likewise Q and P2 are collinear by a line of S. Thus Q E {Pi,P2}. Hence {Pi,P2} = Vb,Ws C 0Ba, and {Pi,P2} = Va,ws C Oab- 24 Now let P\. /N e Oab such that Pi fl 11 v =< A,ws P2 fl 11 -y. = {ri,..., C< Aij2, B > Oab- Thus we see that Oab (and by a similar argument, Ob a) is a pivotal ovoid. One might be tempted to think that just as the packing constructed in the previous section consisted of pivotal spreads, the fan constructed here consists entirely of pivotal ovoids. However this is not always the case. Up until now, no conditions on the oval permutation a were made. At this point, the type of oval we start with will determine certain properties of the associated GQ. Proposition 2.4.3 If < a >= Aut(F), then all ovoids of M. are pivotal. Moreover the trace and span of pairs within one of these ovoids are either affine lines, dual lines in AG(3,q), or planar arcs isomorphic to fi- which complete to hyperovals with A and B. Proof: Oab, Ob a were considered in the previous proposition. Now consider each of the n^. Let X\,X2 enfl6 Pab] let xq =< X\,X2 > PlHoo. If xQ = A, then {.rj. } = {< A,x\,Wi > |w* il } C Oab and {.ri. =< X\,X2 > \{A} C na. Similar results hold when xq = B. 25 Now consider the situation where xQ e< x\,x2 > \{.1. B}. Let x\ = (a, b, c, 1), x2 = (a, e, f, 1). In this case xq = (0, y, 1, 0) where y = (b + e)(c + /)-1 7^ 0 and hence b ^ e. For each ws e there is a unique other point wÂ§ of on < Xq,ws >. Let Ai =< X\,ws >; let A2 =< x2,wÂ§ > and observe Ai flA2 = (a + k, b + ks, c + ksa, 1) = (a + h, e + hs, f + hsa, 1) for some h,k ^ 0. Matching up first coordinates gives h = k. Matching second coordinates gives k = (Â£> + e)(s + s)-1, and matching third coordinates gives k = (c + f)(s + s)~a. From this we have (h+e)(s + s)-1 = (c+f)(s + s)^a implies (s + s) = [(b+e)(c+ Z)-1]13" (we use ^e fact that a generates Aut(F)). Plugging this in above we 1 a 1 get k = (h + e)(s + s)-1 = (h + e)([(h + e)(c + /)-1]T3^)-1 = (6 + e)^3T(c + /)T3^. Now k is written in terms of the coordinates of x\ and x2, hence we can write {.r t} = {(a+k,b+ks,c+ksa,l)\s ef} = T where A: is as given above. It is easy to see that is a g-arc in IIa+fe (note that k is necessarily different from zero). Moreover if we let T1 = U {A, B} we have T1 is projectively equivalent to fi+. To see this let ip be the transformation given a+k b c 1 0 k 0 0 by (t,u,v,w)^ = (t,u,v,w) . Specifically = A, B' = B, 0 0 /c 0 1 0 0 0 and (1, s, sa, 0)^ = (a + k,b + ks, c + ksa, 1). 26 Now let Â£- = {pr = (a, b+(b+e)r, c+(c+/)r, l)|r Â£ F}. For a fixed pr, consider the line formed by pr and some (a+k, b+ks, c+ksa, 1) Â£ {x\,X2}- Such a line would intersect in (1, (b + e)13" (c + /) + s, k^l{c + f )ra + sa, 0). But [(6 + e)13^(c + /)3Tr + s]a = A;_1(c + /)r + sa. This point is on fi- and hence this line is a GQ line. In other words each point of is collinear in the quadrangle with each point of {xi,x2}- Since has size q we see that in fact XT = {xi.jx2}. It is easy to see that X is a q-arc in Ii which forces IIa to be a pivotal ovoid. Note also that X+ = X- U {.1. B\ is projectively equivalent to il . This equivalence is given by a where a b cl (t, u, v, w)a = (t, U, V, tv 0 b + e 0 0 0 0 c+f 0 10 0 0 . Clearly a maps fi+ to X+ with A and B fixed. Therefore each IT* e Va,b is pivotal. Hence M. is a fan with every ovoid pivotal. Examining pairs of points from different ovoids gives the following. Proposition 2.4.4 If all ovoids of M. are pivotal then for x\,x2 in different 27 ovoids, {.r i. } = {.r i. Proof: The proof is similar to that of proposition 2.3.4, with line and spread replaced by point and ovoid. As with proposition 2.3.4, note that the proof depends not on the par- ticular construction but only on the existence of a fan with all members pivotal. This idea of recognizing properties of GQ independent of their constructions will be useful in their classification and characterization. 28 3. Regular ovoids in S In this chapter regular ovoids are examined extensively. The main theme in this chapter is the relationship between regular ovoids and affine planes. Along the way, a method of deriving a GQ of order q from one of order (q + 1, q 1) is presented. Let S = (V, B.X) be a GQ(q + 1, 5 1) with q even. Let Ooo be an ovoid of S, i.e. is a set of 1 + (q + l)(q 1) = q2 points of S, no two on a line. This means that each line of S contains exactly one point of O^. Also assume that is regular: for all .r. // t- {.r. y} is a regular pair and {x^}^ is a set of q points all in O^. 3.1 Affine planes and ovoids This section begins by noting that the points and hyperbolic lines of the regular ovoid O^ can be viewed as the points and lines of an affine plane. The parallel classes of this affine plane are used to construct the remaining ovoids of a fan. Examining these ovoids from the affine point of view will help in determining other characteristics of the GQ. Construct an affine plane 'k{O0O) as follows. Take the q2 points of 29 Oao to be the points of the ^(C^). For x and y in let the hyperbolic line {a;,?/}11 be a line of ,k{0oq). Let incidnce be given by containment. Because 000 is regular, |{rc, 2/}_L_L| = q- Furthermore by observation 1.3.2, each pair of points in 7r(O^ is contained in a unique line of This makes an affine plane (see the appendix). Let /1. /b....l\, be the pairwise disjoint lines of one parallel class of 7r((9oo). The points of these lines account for all the points of the ovoid. Hence Tl U T2 U U Tq = Ocq. The proposition below shows how a related ovoid can be constructed from parallel classes of Proposition 3.1.1 T1 U T\U U T'q is an ovoid of S. Proof: Let I) = {./,. //,} Recall that T1 is the set of points collinear with everything in Tj. Let z\ E Tf1 = {.r 1. //1} and z2 e T^ //_.} If Â£2 Â£ Tj1 then 2/i e T2. (To see this observe that the q points of fl O^ are exactly the q points of T2.) This is a contradiction because Tl is parallel to T2. Thus for all 1 0 j, T)1 fl T1 = 0 which implies IT1 U U T1) = q2. If z[ E Tj1, then zi 0 z[, otherwise a triangle in S is formed. Now suppose z\ ~ Z2; then z\ E {.r 1. ) Since z\ is collinear with each of the q non-collinear points of Tx, 2:2 must be on one of the lines from zi to a point of Tl, say Z2 E y\Z\. Since each of the q lines through Z2 contains exactly one 30 point of T2 = {x2,y2}, the line y\Z\ must be such a line. This means there is a z E y\Z\ such that z 6 T2, giving two points of Ooo,yi and z, which are collinear in S. But this cannot be as is an ovoid. This shows that no two points of Tj1 U U are collinear in S. Therefore Tj1 U U is an ovoid of S. m This shows that for each parallel class of 7r(0oo) there is a corre- sponding ovoid. These ovoids will be of primary interest. However as an aside observe that in fact, many more ovoids can be constructed. Proposition 3.1.2 If T' E {'/}. 1) \. 1 < % < q, then Uf=i T/ is an ovoid. Proof: For distinct i and j, no point of Tt is collinear with a point of Tj. Likewise by proposition 3.1.1, no point of TA is collinear with a point of Tj-. Assume T[ = l\ and = T%- Let //i e T\,z\ E Tf1, Z2 E T%- If E l\. then z2 ~ Z\. As shown above, this cannot happen. Hence Ti CiT,^ = 0, which implies \T[ U U T'q\ = q2. We need to show that no point of Ti is collinear with a point of T^. For contradiction, suppose yi ~ z2. As above, since every line through z2 con- tains a point of there is some z E y\z2 with z ET2. But then z and y\ are two points of Oao which are collinear in S. This contradicts O^ is an ovoid of S. Therefore no two points of Uf=i T' are collinear. 31 Let Eq, Ei, ... Eq he the q +1 parallel classes of lines of 7r(0oo). Then for 0 < i < q, Oi = : T E Ej} is an ovoid as shown above, and M = {Oooi Oo, Oi,..., Oq} is a fan of ovoids. As O^ is regular, it follows that Ooo is pivotal for M. By construction each Oi is partitioned into hyperbolic lines of size q whose traces partition O^, 0 < i < q. These ovoids may or may not also be regular; in some cases they are all regular. The final theorem of chapter 4 will help to classify the regular ovoids. 3.2 Intersection of traces with ovoids of M In this section two propositions from [Pay85a] are reviewed. These propositions indicate how the traces of point pairs intersect the ovoids of M. They will be used in examing regular pairs in Chapter 4 and in characterizing GQ in Chapter 5. Put / = {(). I...(/}:/ = / U {oo}. Proposition 3.2.1 Let b, d E Oj, with j E I,b ^ d. If {b, d}L fi O^ ^ 0 then {M^cOoo. Proof: Let b,d E Oj,b 0 d. Suppose a E {b, d}L fi O^ and c E {b, d}1- with c 0 O^. As each line meets each ovoid of M. exactly once, let a! = be fi CCo and let / be the unique point of ad collinear with a/. As 32 a, a' E Oao (which is pivotal) and b E {a, a'}-1 then all of {a, a'}-1 is contained in Oj\ specifically f E Oj. But d is assumed to be in Oj, a contradiction as d~f. Proposition 3.2.2 Let O^, Oj, Oj be distinct ovoids of M. i) x E Oj, y E Oj, and x ^ y implies |{:c, y}L fi 0,x\ = 1. ii) a E Ooo, b E Oi, and a b implies |{a, C\ Oj| = 1; Proof: For each a E O^, ax contains q(q 1) noncollinear pairs (z,w) E Oi x Oj. As Ooo is pivotal, if a and a' are distinct points in Ooo, then a1- and a'L can never contain the same noncollinear pair (z, w) E OjX Oj. Thus there are (\000\)q(q 1) = q3(q 1) such pairs as a runs over the elements of Oo- On the other hand, counting the number of such pairs shows that there are q2 choices for z E Oj, and q2 q choices for w E Oj with z w. This gives q3(q 1) noncollinear pairs (z,w) E Oj x Oj, which proves part i). For a fixed a E Cl v. .r E Oj, {a^x}1- has exactly q points. If any two of these were in the same ovoid of M then the previous proposition would be contradicted. This proves ii). 33 3.3 Constricting about a regular ovoid At this point additional GQ can be constructed. Having a regular ovoid Ooo in S allows us to form a new GQ, S^, of order q via a method called constriction about 00c. This is done by defining the points, lines and incidence of Lines of Incidence of S^: A line of B is incident with a point of Too provided the two were incident in S. If T is a hyperbolic line of O^, then TL is incident with the q points it contains in S and Tx ~ (Oi) provided Oi is the (unique) ovoid from M. containing TL. Finally is incident with each of the q + 1 points (Oi), 0 < i < q. In chapter 1, it is was noted that the process of expanding about a regular line to form a GQ(q +1, 5 1) from a GQ(q, q) was analogous to forming an affine plane from a projective plane. Similarly, the process of constricting about a regular ovoid in a GQ(q + 1, q 1) to form a GQ(q, q) is analogous to forming a projective plane by adding in a line to an affine plane. 34 3.4 Regularity in from S. Proposition 3.4.1 /, v is a regular line of considered. Let L*, Lj, L& be distinct lines of affine lines in distinct parallel classes of 71(0^). By the dual of observation 1.3.3 it suffices to show that {L,h Lj} is a regular pair. Let L E {Li,Lj}; in S, L intersects O^ in some point x. Hence x is collinear with all of and with all of T^. Let Rx be the set of lines through x thought of as lines of then Rx C {Li,Lj}. In fact as \RX\ = q = { L,. L,} {Loo}|, { /'/ } = Rx U {Loo}. If Lk ~ L then by similar argument L is concurrent everything in /?,: i.e. any line concurrent with the pair {Loo,L} C {Lj^Lj}1- is concurrent with all of {Li,Lj}. From the dual of observation 1.3.1, this means that {Li, Lj} is a regular pair and hence L^ is regular. For the next proposition the following lemma is needed. 35 Lemma 3.4.2 In any GQ{s,t), let L and M be distinct lines through 2 such that each point on L {z} forms a regular pair with each point on M {z}. In this case 2 is itself a regular point. Proof: For any xel {). y e M {} observe that {.r. //} {^}| = t. There are s choices for x and s choices for y, and hence there are s2 traces {x, y}1- containing 2. This accounts for s2t points not collinear with 2. On the other hand, the total number of points of the GQ outside of ^ is (s + l)(st + 1) s(t + 1) 1 = s2t. Now every point not collinear with 2 has been accounted for in these traces. As {x, y} is assumed to be regular, every pair from {x, y}1- must be regular. Specifically 2 forms a regular pair with every point it is not collinear with, i.e. z is regular. Thus any noncollinear pair in is regular. We will now consider the relationship between regular points in and pivotal ovoids in S. Proposition 3.4.3 (Oj) is a regular point of Proof: 1) {{Oj) is regular implies Oj is pivotal for M) In what follows assume that x and y are points of S^ which were also points of S; i.e. they are not points of the form {Oj). 36 {x,y} is a set of non-collinear points of (0,i)L in if and only if x,y E Oi and x and y are in distinct traces, Tf1, of the hyperbolic lines, say '/'i and T2, of C> v such that Tf1, C C>( and Ti fi T2 = 0. (C?i) is a regular point of implies {x, y} is a regular pair in S^, and hence {.r. //. } = {x\ = x,X2 = //. x:>. x\.....rq. i}. Each point of {.r. //} is collinear with each point of {x,y}. Each line through (Oi) of the form Tf- contains a unique point of {:c, t/}-11. As there are only q such lines, the remaining point must be some (Oj). This gives {x, y}1- = {t/i, t/2,..., yq, yq+\ = (Oi)} and {./. y}^ = {.r,. .....rq. xq+i}. Thus |/i, y-2, , yq are all collinear with (Oj) in point of S0o, any two points contained in Oi (in S) have their trace in an ovoid Oj of M and have their span completely contained in Oj. This means Ot is pivotal for M. 2) (Oi is pivotal for M in S implies (Oi) is a regular point of Soo.) Assume Ot is pivotal for M. Let x1;x2 be collinear with (Oi) such that Xi 9^ x2 and x\,x2 not on L v. Then x\,x2 e Oi, and if Oi = T-j1 U -UT^ with Ooo = Ti U U Tq, without loss of generality assume x\ E I\.x2 E T2. In S, since Oi is pivotal, {.r t} = {.r t.....rq} C Oi and {.r t} = {yi,...,yq} C Oj for some Oj E M. Then for 1 < i,j < q, xt ~ yj 37 in S and hence also in S^. Back in in S0c. By lemma 3.4.2, {aq, (Ok)} is regular for all (Ok) on L^. Therefore (Oi) is a regular point of S^. In addition to the relationship between regular points of and piv- otal ovoids in S, there is also an interesting relationship between planes asso- ciated with Soo and planes associated with S. Here, too, regularity plays a key role. For the moment, consider the plane ^(Ooo). Let Â£0 = { /'i 7L>...., Tq} and E\ = {i?i, R2,..., Rq} be two distinct parallel classes of lines in ^(O^). Each point of ,k(000) is on a unique hyperbolic line Tj and a unique hyperbolic line Rj, hence these points may be labeled as aqj = Tj fi Rj. Tj- _ Xi^}1- for any h ^ j. and Rj- = {aq(. ,r/iV} for any k ^ i. In what follows the notation is abused somewhat, with the _L notation used for whichever GQ is convenient. The context should make it clear what the notation means. To aid in this, when Tj- and Rj- are viewed as lines of S^, we will write I., = Xj- and .1 /( = Rj-. Traces and spans of lines in can be determined as follows. In the GQ the line L00. Every line of {,l/(} can be viewed as the trace in S of a hyperbolic line of which contains Xij. From the regularity of L00 in S^, there is an affine plane tt(L(*,) which arises as in construction 1.3.10. The point set here can be viewed as Mj\ : 1 < i,j < q}; Â£(Too) = {HL : H is a hyperbolic line of C^}. In this setting, if Hx is an affine line of ,k(L00), the affine points incident with H1- are the hyperbolic points in Proof: Deline ip from V(Ox.) to V(L0c) by 'tp(xij) = { !) Hj } . To see that ip preserves col linearity, first suppose that l = Tt = {xiA...., x,^q}. In which case 'tp(l) = {{1) /t^ } . {1) ./?._;} ..{l) Iiq } i.e.. ex- actly the set of points in P>(T00) on the line Tp-. Now suppose that l ^ Â£o, say l = {xhri,x2,r2, ,%rj- m is then {ip(xhri), -ip(x2,r2), , ^(%rg)} which equals { { /', } {7b . } ...., {Tq /i>(: } } which in turn is equal to {{ /. i^}^, {7b . ___{7 / } }. But this is exactly the set of points in 'P(Too) on the line lL E Â£(Â£oo). Thus ip maps lines to lines. Thus ip is an isomorphism between ^(0^) and 39 Now suppose that the ovoid Oq is also pivotal for the fan M. so that the point {Oq) of L00 is regular in Soq. From the regularity of the point {Oq) in there is an affine plane 7r( {Oq) ) which arises as in construction 1.3.8. This new affine plane has point set V{ {Oq) ) = {x : x e Oq} and line set Â£( {Oq) ) = {{rr, t/}JJ : x,y e Oq}. But these are exactly the point- and line sets of k{Oq), and the incidences are the same. This proves the following theorem. Theorem 3.4.5 The two affine planes 7r( {Oq) ) and k{Oq) are identical (not merely isomorphic). In the case under consideration here, i.e., that both (D^ and Oq are pivotal for M. in S, Payne has shown in [Pay72a], [Pay77] and Chapter 12 of [PT84] that consequence Theorem 3.4.6 This theorem will be utilized in a characterization given later. 40 3.5 Grid-like Fans Next consider the associated planes in the case where at least three of the ovoids of M. are pivotal. To answer the question, Are these planes isomorphic, a new axiom is introduced and examined. Let S be a GQ(q + 1 ,q 1) with a fan M. Grid-Like Axiom: Let Ot, Oj, Ok be any triple of distinct pivotal ovoids for M with ./ i. E O,. i/i, |/2 E Oj, z\,z2 E Ok such that {xi,x2} then every line meeting {.rj. ) and {2/i,2/2} also meets {zi,z2}- Any fan with at least pivotal ovoids satisfying this axiom will be called a grid-like fan. In section 2.4, P(T2(Q), < A,B >) was shown to have a fan M. = {Oab, Oba}UVa,b with every ovoid pivotal provided fi was a translation oval, i.e. < a >= Aut(F). Additionally, the following theorem holds. Theorem 3.5.1 M. = {Oab, Ob a} U Va,b is a grid-like fan. Proof: Let I\. I2 E Oab, and let Qi,Q2 E Oba Let xi,x2 E na, 2/1,2/2 ^ na+m, Z\, z2 E na+n with m, n, 0 all distinct such that X\, 2/1, Z\, P\, and Qi are all collinear in S by Ai, and x2,y2, z2, P2,Q2 are all collinear in S by A-_>. P\ =< 1. A1 2>. (J\ =< B, Ai ^>,P2 =< 1. A_> >, and (J2 =< B. A_> >. Let Xq =< Xi,x2 > nll.v 2/0 =< 2/1, y2 > rll v A =< Zi,z2 > miv. Let 41 Ws Cl II x. A2 0 Iloo Construct the hyperbolix lines H\ = {.r i. } . //_. = {//1. //_,} //.> = {-i- c} //. = {/C/A} and //- = . Let H = {//,...........//-,}. To show that M. is grid-like, we need to show that any line intersecting two hyperbolic lines of % must intersect every line of %. To do this, we must con- sider the various ways xQ,yQ, and zQ could be related to each other and to A and B. The following lemma will assist in this. Lemma 3.5.2 If x0 ^ yo, then in fact xQ ^ zQ ^ y0. Proof of lemma: Suppose for contradiction that t/0 = %o- Let II* be the plane < t/o, Vi, > Observe the following implications. z\,y\ E II* implies Xi,ws E 11,: Z\, Zq E 1I* implies z2 E 11,,,: z2, //_> C 11* implies x2, wt E 11,: xi,x2 E II* implies xQ E II*. Finally xQ,yQ E II* implies A,B E II*. Thus Iloo fi II* is line containing A, B,ws,wt, a contradiction. Hence r/o ^ zq, and similarly xQ ^ zQ. This proves the lemma, and return now to the proof of the theorem. Consider the following cases and subcases for x0,yo, and z0: (1) xq = A with (a) x0 = y0 = z0, or (b) xo,yo, zq are distinct members of < A, B >. 42 (2) xq = B (But results here will be like those above with A and B inter- changed.) (3) xQ = y0 = zQ G< A, B > \{. 1. B} with (a) ws = wt, or (b) ws wt. Case 1: xQ = A. Let x\ = (a. b. c. I )../_. = (a,b + e,c,l),iJi = (a + m,b + ms, c + msa, 1), t/2 = (a + m, b + e + mt, c + mta, l),z\ = (a + n,b + ns, c + nsa, 1), 1J2 = (a + n,b + e + nt,c + nta, 1). First consider A = xQ = y0 = zQ (this is true if and only if s = t). If x?j G {xi, X2},'i/3 G {yi,'tj2} with xq ~ y$, then there exists some d G F such that Xq = (a,b + d, c, 1) and y3 = (a + m,b + d + ms, c + msa, 1). Let Zq = (a + n,b + d + ns, c + nsa, 1). Observe that z3 G {i. z2} and z3 G< xq,ws >=< Xq, i)Q >. This means that any S line meeting {.r t} and {yuU2} must also meet {z1,z2}. Also, observe that Pi = P2 G {xi,x2}, and that < B,XQ,yQ >G {Qi,Q2}- Now consider xq = A,y0,ZQ as three distinct members of < A,B >, and let x\. yx. //_,.i. be as above. Without loss of generality, consider l/o = B, Then y2 = (a + m,b + ms, c + msa + h, 1) for some h. Comparing this to y2 as written above observe t = em1 + s and h = ml^aea. So in fact 1/2 = (a + m, b + ms, c + msa +ml^aea, 1), and also z2 = (a + n, b + e, n{emTl + 43 s), c + n(em-1 + s), 1). Again choose xz E {%i, X2},V3 Â£ {yi,V2} with xz ~ yz- Thus Xz = (a, b + d, c,l),yz = (a + m,b + ms, c + msa + /i, 1) such that < .r3. |/3 > intersects Lt^ at a point (m, d + ms, /i + ms, 0) E fi+ Dividing through by m gives (dmrl+s)a = sa+hmrl which forces h = dam1-a. In fact y3 = (a+m, b+ ms, c + msa + dam1-a, 1), and < Xz, yz > Cin^ = (1, s + dm-1, sa + dam~a, 0). From this it follows that < .r3. y3 > filla+n = (a + n,b + d + n(s + dm-1), c + n(sa + dam~a), 1). Call this point z3. Now observe that z3 E {zi,z2} = {(a + n, b + ns, c + nsa, 1), (a + n, b + ns + (1 + nm-1)e, c + nsa + nm~aea, 1)}J~ = {(a + n, b + ns + (1 + nm-1)er,c + nsa + nm-aeara, l)|r E F} (to see this let r = de-1). Any Saline meeting {.rt) and {yi,y2} must also meet {+, z2}- Observe that < A,xz,yz >e {Pi,P2} and < B,x3,yz >e {Qi,Q2}- As was noted above, case 2 is similar to case 1. Case 3: xq = y3 = zq = (0, //. 1.0). // ^ 0. Let x\ = (a, b, c, I). x2 = (a, e, f, 1) which forces y = (b + e)(c+/)_1. Let y\ = (a + m, b + ms, c + msa, 1). Consider ws = wy, thus y2 = (a + m, e + ms, f + msa, 1), Z\ = (a + n, b + ns, c + nsa, 1), and z2 = (a + n, e + ns, / + ns, 1). Again let x3 e {.ri. } . y3 e {di, y2}J- with x3 ~ y3. Xz = (a,b+(b + e)r, c + (c + /)r, 1), r ^ 0,1; y3 = (a + m,b + ms + (6 + e)t>, c + ms + (c + /)f, 1) for some v. 44 x3 ~ 1/3 implies (1, s+m~l{b+e){r+v), sa+mr1(c+f)(r+v)a, 0) e which in turn implies that [s + m^l{b + e)(r +1>)] = sa + m~l{c + f)(r + v)a. Note that if r ^ r then rri = (c + f)y'x-T in which case < X2,yi > rffloo = 1 cx (1 /y1^ + s/y1^ + s,0) e But this means that x2 ~ y\ and therefore yi e {xi,x2\l. The only case needing to be considered is when r = v. But in this case < yz > nrioo = ws, and hence < x:>. y2j > nn3 = (a + n, b + ns + (b + e)r, c + nsa + (c + f)ra, 1) e {zi, z2}- Once again, any S line meeting {.rt} and {|/i, y2} must also meet {z\, z2}- Also note that < A, x?>. y3 > = < A,xz,ws >G {Pi,P2} and likewise < B,x$,y$ >e {Qi,Q2}- Finally consider ws ^ wt. Since wt G< xQ,X\,yi > Hi 1 v. then , 1 a *. 1 wt G< ws, x0 >. Thus wt = (I. k //1 . sn //1 . 0). i.e, t = s + y1- . 1 cx From this we see thatt/2 = (a + m,e + ms + my, / + msa + my, 1), z\ = . , . 1 cx , (a + n,b + ns, c + nsa, 1), z2 = (a + n,e + ns + ny1-**, j + nsa + ny1-a 1). As before, let x$ = (a, b + (b + e)r, c + (c + f)ra, 1), r ^ 0,1. If yz is again chosen to be the point of {t/i, t/2}JJ which is on an S line with x$ then 1 cx y?> = (a + m,b + ms + (b + e + m,yT^)r, c + msa(c + / + m,yT^)ra, 1) and 1 < Â£3,2/3 > intersects IIoo at the point wu where u = s + yl-ar. 1 cx Thus z3 = (a+n,b+ns+(b+e+nyI^)r,c+nsa(c+f+nyI^)ra,l) =< x3,y3 > n{^2}. Yet again (as has been the refrain) any S line meeting 45 {.ri. and {yi,'tj2} must also meet {zi,z2}- To see that < x^,y^ > intersects {Pi, P2}, first observe that the point t>3 = (a + (c + fyy^1, (c + f)y'^-Ts,c+ (c + f)y'^-Tsa, 1) is a point of Pi fi P2, and thus Pi =< A, v3, ws > ,P2 =< A. r3. ir, >. From this we see that {Pi. P2} is the set of planes through the line < A, t>3 >. Finally observe that t>3 E P3 =< 1. .r3. n\, > and P3 E {PL,P2}. Similarly, < x3,|/3 > is contained in a plane of {Qi,Q2}- Hence M is grid-like. Now that the concept of grid-like is seen to be non-vacuous, the axiom is used to prove a theorem about the associated affine planes. Return to the notation of letting S be any GQ(q + 1,5 1) with a fan M. If any ovoid of M is pivotal then in fact M must have been constructed as in section 3.1. To see this first note the following lemma. Lemma 3.5.3 If Cl v is pivotal for M and Â£0 = {Ti,.. .,Tq} is a parallel class of the associated affine plane 'k{O0O).j then Tf1 U ... U Tq is an ovoid of M. Proof: doo is pivotal implies Tf1 C d* for some d* M. Let x E Tj ^ Ti. Choose y E xL fi d* and choose x' E yL fl doo\{a;}. Suppose z E {./. PjT-1 fl T\. Then z ~ y implies y is in Tf1 and x is in I\. a contradic- tion. Therefore {./../'} E t\,. i.e. {./../'} = 7). As Tj1 fl dj is not empty, Tj- must be contained in dj. 46 With this proved, it follows that given any parallel class of ^(Ooo) the traces of the lines in the parallel class partition the points of an ovoid of M. Because the number of ovoids in equals the number of parallel classes in ^(Ooo), each ovoid of Ad\{(Aoo} can be viewed as the union of traces of hyperbolic lines in a given parallel class of 71(0^). This proves the following theorem. Theorem 3.5.4 Every regular ovoid of S is pivotal for exactly one fan, namely the fan constructed in the manner of Section 3.1. We are now in a position to use the Grid-like Axiom to identify iso- morphic planes. Theorem 3.5.5 If M. is a grid-like fan, then the affine planes associated with the pivotal ovoids of M. are all isomorphic. Proof: Assume that M. is grid-like. Let O^.Oq.Oi be distinct pivotal ovoids in M. Let Â£q,Â£i be parallel classes of 7r((Aoo) such that each hyperbolic line of Â£0 has its trace in O0 and each hyperbolic line of Â£\ has its trace in 0\. Furthermore let JF0 = {T^jT e Â£0} and T\ = { R A E Â£{\. Label the members of Â£q by Â£q = {I\. '/_...., Tq} and let R\, i?2 be distinct members of Â£\. Identify particular points of Clx.. For i = 1,2 let = I) n R,. 47 {.ri. meets each member of Â£Q and each member of E\. For i = 3............q let Xi = Ti fl {.ri. } and let Ri be the unique line of E\ containing x%. For each i = 1,..., q let {A^i,... Aij(?} be the set of lines from S which are incident with Xi and let = A*j fl Oq, bi;j = A*j fl 0\. Dehne the map iai,j) = h,j- The claim is that morphism from 7t( It remains to be shown that
7,(0,). fore
Corollary 3.5.6 If M is a grid-like fan, then either all or none of the affine |

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