Springer Monographs in Mathematics More information about this series at http://www.springer.com/series/3733 J.W.P. Hirschfeld • J.A. Thas

General Galois Geometries J.W.P. Hirschfeld J.A. Thas Department of Mathematics Department of Mathematics University of Sussex Ghent University Brighton, UK Gent, Belgium

ISSN 1439-7382 ISSN2 196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-1-4471-6788-4 ISBN 978-1-4471-6790-7 (eBook) DOI 10.1007/978-1-4471-6790-7

Library of Congress Control Number: 2016930023

Mathematics Subject Classification (2010): 51E20, 51E12, 51E14, 51E22, 51E23, 94B05, 05B25, 14H50, 14L35, 14M15, 51A05, 51A30, 51A45, 51A50, 51D20, 51E21

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Contents

Preface ...... xi

Terminology ...... xv

1 Quadrics ...... 1 1.1 Canonical forms ...... 1 1.2 Invariants...... 3 1.3 Tangencyandpolarity...... 7 1.4 Generators...... 13 1.5 Numbers of subspaces on a quadric ...... 19 1.6 The orthogonal groups ...... 21 1.7 Thepolarityreconsidered...... 29 1.8 Sections of non-singular quadrics ...... 31 1.9 Parabolicsectionsofparabolicquadrics...... 39 1.10Thecharacterisationofquadrics...... 42 1.11Furthercharacterisationsofquadrics...... 51 1.12 The Principle of Triality ...... 53 1.13 Generalised hexagons ...... 55 1.14Notesandreferences...... 56

2 Hermitian varieties ...... 57 2.1 Introduction ...... 57 2.2 Tangencyandpolarity...... 58 2.3 Generatorsandsub-generators...... 64 2.4 Sections of Un ...... 65 2.5 The characterisation of Hermitian varieties ...... 69 2.6 Thecharacterisationofprojectionsofquadrics...... 80 2.7 Notesandreferences...... 96

vii viii Contents

3 Grassmann varieties ...... 99 3.1 Pl¨uckerandGrassmanncoordinates...... 99 3.2 Grassmannvarieties...... 107 3.3 AcharacterisationofGrassmannvarieties...... 121 3.4 EmbeddingofGrassmannspaces...... 137 3.5 Notesandreferences...... 142

4 Veronese and Segre varieties...... 143 4.1 Veronesevarieties...... 143 4.2 Characterisations...... 153 V2n 4.2.1 Characterisations of n ofthefirstkind...... 153 V2n 4.2.2 Characterisations of n ofthesecondkind...... 163 V2n 4.2.3 Characterisations of n ofthethirdkind...... 180 V2n 4.2.4 Characterisations of n ofthefourthkind...... 181 4.3 Hermitian Veroneseans ...... 196 4.4 Characterisations of Hermitian Veroneseans ...... 198 4.4.1 Characterisations of Hn,n2+2n ofthefirstkind ...... 198 4.4.2 Characterisation of Hn,n2+2n ofthethirdkind...... 199 4.4.3 Characterisation of H2,8 ofthefourthkind...... 200 4.5 Segrevarieties...... 201 4.6 Regular n-spreads and Segre varieties S1;n ...... 212 4.6.1 Construction method for n-spreads of PG(2n +1,q) ...... 219 4.7 Notesandreferences...... 219

5 Embedded geometries ...... 223 5.1 Polarspaces...... 223 5.2 Generalisedquadrangles ...... 226 5.3 Embedded Shult spaces ...... 232 5.4 LaxandpolarisedembeddingsofShultspaces...... 247 5.5 Characterisationsoftheclassicalgeneralisedquadrangles...... 253 5.6 Partialgeometries...... 266 5.7 Embeddedpartialgeometries...... 269 5.8 (0,α)-geometriesandsemi-partialgeometries ...... 272 5.9 Embedded (0,α)-geometriesandsemi-partialgeometries...... 281 5.10Notesandreferences...... 299

6 Arcs and caps ...... 305 6.1 Introduction ...... 305 6.2 Capsandcodes...... 307 6.3 The maximum size of a cap for q odd...... 314 6.4 The maximum size of a cap for q even...... 319 6.5 General properties of k-arcsandnormalrationalcurves...... 325 6.6 The maximum size of an arc and the characterisation of such arcs . . 332 6.7 Arcs and hypersurfaces ...... 338 6.8 NotesandReferences...... 360 Contents ix

7 Ovoids, spreads and m-systems of finite classical polar spaces ...... 363 7.1 Finiteclassicalpolarspaces...... 363 7.2 Ovoidsandspreadsoffiniteclassicalpolarspaces...... 364 7.3 Existenceofovoids...... 365 7.4 Existenceofspreads...... 365 7.5 Openproblems...... 366 7.6 m-systems and partial m-systems of finite classical polar spaces . . . 367 7.7 Intersections with hyperplanes and generators ...... 368 7.8 Bounds on partial m-systems and non-existence of m-systems..... 369 7.9 m-systems arising from a given m-system...... 372 7.10 m-systems, strongly regular graphs and linear projective two-weightcodes...... 374 7.11 m-systemsandmaximalarcs...... 376 7.12 Partial m-systems,BLT-setsandsetswiththeBLT-property...... 378 7.13 m-systemsandSPG-reguli...... 380 7.14Smallcases...... 382 7.15Notesandreferences...... 383

References ...... 387

Index ...... 405

Preface

This book is the second edition of the third and last volume of a treatise on projec- tive spaces over a finite field, also known as Galois geometries. The first volume, Projective Geometries over Finite Fields (1979, 1998), with the second edition re- ferred to as PGOFF2, consists of Parts I to III and contains Chapters 1 to 14 and Appendices I and II. The second volume, Finite Projective Spaces of Three Dimen- sions (1985), referred to as FPSOTD, consists of Part IV and contains Chapters 15 to 21 and Appendices III to V. The present volume comprises Part V and, in its first edition, contains Chapters 22 to 27 and Appendices VI and VII. In this edition, the chapters are numbered from 1 to 7. The scheme of the treatise is indicated by the titles of the parts: Part I Introduction Part II Elementary general properties Part III The line and the plane Part IV PG(3,q) Part V PG(n, q) There are three themes within the book: (a) properties of algebraic varieties over a finite field; (b) the determination of various constants arising from the combina- torics of Galois spaces such as the maximum number of points of a subset under certain linear independence conditions; (c) the identification in Galois spaces of var- ious incidence structures. Many of the results on theme (a) could be equally well stated over an arbitrary field. However, over a finite field, counting arguments come more into play. A significant number of theorems count certain sets and establish the existence of combinatorial structures. Most of Chapters 1 to 4 is on theme (a), whereas Chapter 5 is on theme (c) and Chapter 6 is for the most part on (b). Chapter 7isonthemes(a)and(b). Chapter 1 on quadrics develops their properties and gives one way of character- ising them. Chapter 2 on Hermitian varieties similarly develops their properties and charac- terises them in the course of describing all sets of type (1,r,q+1). This chapter is

xi xii Preface the one on algebraic varieties that differs most from the classical case, as Hermitian manifolds over the complex numbers are not algebraic varieties. Chapter 3 on Grassmann varieties and Chapter 4 on Veronese and Segre varieties most closely follow a classical model in the description of their properties. Although most of the characterisations of the Veronesean of quadrics and its projections re- semble classical theorems over the complex numbers, the characterisation of Grass- mannians is quite different. This is because the Grassmannian characterisation is in terms of an incidence structure, a topic which was studied over the real and complex numbers only for the entire rather than any substructure, whereas the Veronesean and its projections are studied as subsets of PG(n, q) in terms of sections by subspaces. Chapter 4 also contains a section on Hermitian Veroneseans; this section contains no proofs. Chapter 5 begins with polar spaces, thereby unifying the subjects of Chapters 1 and 2, and it goes on to consider the special case of generalised quadrangles and structures which are natural developments. In this chapter, not every theorem is proved; in particular, no proofs are given for most of the characterisations of gen- eralised quadrangles Chapter 6 generalises to an arbitrary dimension results of Chapters 18 and 21 from the previous volume: an upper bound is found for the size of a k-cap and the maximum size of a k-arc is found under some restrictions on n and q; the corre- sponding arcs are generally normal rational curves. Chapter 7 begins with ovoids and spreads of finite classical polar spaces, which are then generalised to m-systems. Applications to maximal arcs, translation planes, strongly regular graphs, linear codes, generalised quadrangles and semi-partial ge- ometries are given. This is the only chapter without proofs. The book is conceived as a work of reference and does not have any exercises. However, each individual chapter is suitable for a course of lectures. Apart from Chapter 5 and the short Chapter 7, complete proofs are given for nearly all results. The last section of each chapter contains all references as well as remarks both on the chapter itself and on related aspects that are not covered. This volume may be considered as developing over finite fields aspects of the three volumes of Hodge and Pedoe [183, 184, 185], particularly regarding quadrics and Grassmannians. Burau [60] is also an appropriate analogy for quadrics, Grass- mannians, Veroneseans and Segre varieties. Compared to the first edition, this edition contains a considerable amount of new material. In Chapter 4, the characterisation of quadric Veroneseans has been com- pletely rewritten; there is also a section on Hermitian Veroneseans. Chapters 5 and 6 are updated, and contain several new and better proofs. Chapter 7 is new up to the section on ovoids and spreads of finite classical polar spaces, and covers much new material but without proofs.

Status of the subject

Apart from being an interesting and exciting area in combinatorics with beautiful results, Galois geometries have many applications to coding theory, algebraic geom- Preface xiii etry, , design theory, graph theory, cryptology and group theory. As an example, the theory of linear maximum distance separable codes (MDS codes) is equivalent to the theory of arcs in PG(n, q); so all results of Sections 6.5 to 6.7 can be expressed in terms of linear MDS codes. Finite is essential for finite , and finite algebraic curves are used to construct interesting classes of codes, the Goppa codes, now also known as algebraic-geometry codes. Many interesting incidence structures and graphs are constructed from finite Hermitian varieties, finite quadrics, finite Grassmannians and finite normal rational curves. Further, most of the objects studied in this book have an interesting group; the classical groups and other finite simple groups appear in this way. Currently there are several international journals on combinatorics and geometry publishing a large number of papers on Galois geometries; for example, Ars Com- binatoria, Combinatorica, Designs, Codes and Cryptography, European Journal of Combinatorics, Finite Fields and their Applications, Journal of Algebraic Combina- torics, Journal of Combinatorial Designs, Journal of Combinatorial Theory Series A, Journal of Geometry, and the conference series Annals of Discrete Mathematics. Finite vector spaces and hence also finite projective spaces are of great impor- tance for theoretical computer science. So, in many syllabuses of a computer science degree, there is a course on discrete mathematics with a section on combinatorial structures.

Related topics

There are some interesting topics either not covered or only touched upon in the three volumes. In the Handbook of Incidence Geometry [55], edited by Buekenhout, surveys of several of these topics are given. Recent surveys are contained in Current Research Topics in Galois Geometry [298], edited by Storme and De Beule. Finite non-Desarguesian planes are not discussed in the treatise. For references see the chapters in the Handbook [55] on ‘Projective planes’ by Beutelspacher and ‘Translation planes’ by Kallaher. See also the book Foundations of Translation Planes [28] by Biliotti, Jha and Johnson and the Handbook of Finite Translation Planes [187], by Johnson, Jha and Biliotti. Spreads and partial spreads in PG(n, q) are considered in Chapter 4 of PGOFF2, in Chapter 17 of FPSOTD, and in Chapter 7 here. For blocking sets, only the plane case is considered in Chapter 13 of PGOFF2. For the theory of spreads, partial spreads and blocking sets in n dimensions, see Sections 7 and 8 of the chapter ‘Pro- jective geometry over a finite field’ by Thas in [55], as well as Chapter 2 by De Beule, Klein and Metsch and Chapter 3 by Blokhuis, Sziklai and Sz˝onyi in [298]. Flocks of quadrics in PG(3,q) are key objects for the constructions of some new classes of translation planes and generalised quadrangles. They also have other applications. For literature on flocks, see Chapter 7 by Thas in [55], and the books Translation Generalized Quadrangles [352] by Thas, K. Thas and Van Maldeghem and Finite Generalized Quadrangles [260] by Payne and Thas. xiv Preface

Ovals and ovoids can be generalised by replacing their points with m-dimensional subspaces. These have connections to generalised quadrangles, projective planes, cir- cle geometries, flocks, and other structures; see the last two books. In Chapter 5, the finite classical generalised quadrangles are considered. Gener- alised quadrangles are the polar spaces of rank 2, the point of view of Chapter 5, but are also the generalised n-gons with n =4. Generalised 6-gons or hexagons ap- pear in Chapter 1. Standard works on generalised n-gons are the books Generalized Polygons [391] by Van Maldeghem and Moufang Polygons [385]byTitsandWeiss. Although null polarities are mentioned in Chapter 7, they are not discussed in detail, nor are pseudo-polarities; references are given there. The book contains only a few group-theoretical results; also theorems on graphs and designs are rare. Apart from the Handbook of Incidence Geometry, the books of Dembowski [116], Beth, Jungnickel and Lenz [18, 20, 19], Brouwer, Cohen and Neumaier [41], Cameron and van Lint [65], Hughes and Piper [186], Assmus and Key [4] may be consulted. Much material is contained in the Handbook of Combi- natorial Designs [73], edited by Colbourn and Dinitz. Reference works on point-line incidence structures and diagram geometries are Diagram Geometries [253] by Pasini, Points and Lines [288] by Shult, Diagram Geometry [56] by Buekenhout and Cohen. Codes are considered only in Section 2 of Chapter 6. For an introduction, see Hill [170] or van Lint [388]. For further results and geometrical connections, see Cameron and van Lint [65], MacWilliams and Sloane [222], Peterson and Weldon [264], Tonchev [386]. For an introduction to Goppa’s algebraic-geometry codes, see Pretzel [267], van Lint and van der Geer [389], Goppa [144], Moreno [240], Nieder- reiter and Xing [242], Hirschfeld, Korchm´aros and Torres [176]. For a range of other topics, see the Handbook of Finite Fields [241], edited by Mullen and Panario.

Acknowledgements

Initial versions of Chapters 4 and 7 were typed by Sonia Surmont, of Chapter 5 by Annelies Baeyens, and of Chapter 6 by Erin Pichanick. We are extremely grateful to them for their diligence. The authors also thank the University of Sussex and Ghent University. Both in- stitutions have supported the writing of this book The first author is grateful to Professor T.G. Room and Professor W.L. Edge, the supervisors of his MSc and PhD, who started him on this subject. The second author is also very grateful to Professor J. Bilo, who initiated him into the mysteries of classical geometry thereby providing him with the necessary background to appreciate the beauty of Galois geometry. Brighton and Ghent JWPH 25 August 2015 JAT Terminology

The set V (n+1,K) is (n+1)-dimensional over the field K and is taken to be the set of vectors X =(x0,...,xn),xi ∈ K. Correspondingly, PG(n, K) is n-dimensional projective space over K and is the set of elements, called points, P(x) with x ∈ V (n +1,K)\{0}.WhenK =GF(q)=Fq, the finite field of q elements, also called the Galois field of q elements, then V (n+1,K) is written V (n+1,q) and PG(n, K) is written PG(n, q).Theorder of PG(n, q) is q. The number of points in PG(n, q) is qn+1 − 1 θ(n)= . q − 1

A projectivity,orprojective transformation, from S1 to S2, with S1,S2 both n-dimensional projective spaces over Fq, is a mapping T : S1 → S2 such that P(x)T = P(xT ) for all vectors x =0 and some non-singular (n+1)×(n+1) matrix T . The group of projectivities from PG(n, q) to itself is denoted PGL(n +1,q).A collineation from S1 to S2 is a mapping T : S1 → S2 preserving the incidence of points and lines. The Fundamental Theorem of Projective Geometry states that σ P(x)T = P(x T ) with σ an automorphism of Fq. Mostly, the properties considered are invariant under PGL(n +1,q). A reciprocity of PG(n, q) is a collineation T from PG(n, q) to its dual space; if T is a projectivity, then the reciprocity is a correlation of PG(n, q). A subspace of dimension r in PG(n, q) is a PG(r, q) and is written Πr;this notation is used both specifically and generically. Then Π−1 is the empty set, Π0 is a point, Π1 is a line, Π2 is a plane, Π3 is a solid, Πn−1 is a hyperplane.Also,π(u), with u =(u0,...,un), with not all ui zero, denotes the hyperplane whose points P(x0,...,xn) satisfy the equation

u0x0 + ···+ unxn =0.

A subspace written πr can have any dimension. In PG(n, q), the vertices of the simplex of reference are denoted U0, U1,...,Un,whereUi has 1 in the (i +1)-th coordinate place and zeros elsewhere, and U is the unit point. Dually, u0, u1,...,un

xv xvi Terminology are the hyperplane faces of the simplex of reference and u is the unit hyperplane. The set of all r-spaces in PG(n, q) is written PG(r)(n, q). If two subspaces S, S intersect in a point P , this will generally be written

S∩S = P.

∗   For any matrix M =(mij ),thetranspose M =(mij ) has mij = mji. The ring Γ=Fq[X0,...,Xn] is the ring of polynomials in the indeterminates X0,...,Xn over Fq.ForF1,...,Fr non-zero forms, or homogeneous polynomials, in Γ,thevariety

V(F1,...,Fr)={P(x) ∈ PG(n, q) | F1(x)=···= Fr(x)=0}.

So the hyperplane π(u) is also written as

V(u0X0 + ···+ unXn).

The term ‘variety’ here is the set of rational points of a variety in the sense of alge- braic geometry. A variety V(F ) is called a hypersurface. A hypersurface in PG(2,q) is a plane algebraic curve; a hypersurface in PG(3,q) is a surface. If the hypersur- faces F1 and F2 are projectively equivalent, then write F1 ∼F2. In keeping with the terminology of Chapter 8 of PGOFF2, in PG(2,q)anoval is a (q +1)-arc for q odd and a (q +2)-arc for q even. Other authors use hyperoval or complete in the latter case. Occasionally, (r, s) denotes the greatest common divisor of r and s. For more detailed explanation of the foregoing, see Chapter 2 of PGOFF2. PART V

PG(n, q)