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E-raamat: General Galois Geometries

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This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of plane case (first volume) and three dimensions (second volume). 

This revised edition includes much updating and new material. It is a mostly self-contained study of classical varieties over a finite field, related incidence structures and particular point sets in finite n-dimensional projective spaces. 

General Galois Geometries is suitable for PhD students and researchers in combinatorics and geometry. The separate chapters can be used for courses at postgraduate level.
Preface xi
Terminology xv
1 Quadrics
1(56)
1.1 Canonical forms
1(2)
1.2 Invariants
3(4)
1.3 Tangency and polarity
7(6)
1.4 Generators
13(6)
1.5 Numbers of subspaces on a quadric
19(2)
1.6 The orthogonal groups
21(8)
1.7 The polarity reconsidered
29(2)
1.8 Sections of non-singular quadrics
31(8)
1.9 Parabolic sections of parabolic quadrics
39(3)
1.10 The characterisation of quadrics
42(9)
1.11 Further characterisations of quadrics
51(2)
1.12 The Principle of Triality
53(2)
1.13 Generalised hexagons
55(1)
1.14 Notes and references
56(1)
2 Hermitian varieties
57(42)
2.1 Introduction
57(1)
2.2 Tangency and polarity
58(6)
2.3 Generators and sub-generators
64(1)
2.4 Sections of Un
65(4)
2.5 The characterisation of Hermitian varieties
69(11)
2.6 The characterisation of projections of quadrics
80(16)
2.7 Notes and references
96(3)
3 Grassmann varieties
99(44)
3.1 Plucker and Grassmann coordinates
99(8)
3.2 Grassmann varieties
107(14)
3.3 A characterisation of Grassmann varieties
121(16)
3.4 Embedding of Grassmann spaces
137(5)
3.5 Notes and references
142(1)
4 Veronese and Segre varieties
143(80)
4.1 Veronese varieties
143(10)
4.2 Characterisations
153(43)
4.2.1 Characterisations of V2n of the first kind
153(10)
4.2.2 Characterisations of V2n of the second kind
163(17)
4.2.3 Characterisations of V2n of the third kind
180(1)
4.2.4 Characterisations of V2n of the fourth kind
181(15)
4.3 Hermitian Veroneseans
196(2)
4.4 Characterisations of Hermitian Veroneseans
198(3)
4.4.1 Characterisations of Hn,n2+2n of the first kind
198(1)
4.4.2 Characterisation of Hn,n2+2n of the third kind
199(1)
4.4.3 Characterisation of H2,8 of the fourth kind
200(1)
4.5 Segre varieties
201(11)
4.6 Regular n-spreads and Segre varieties s1;n
212(7)
4.6.1 Construction method for n-spreads of PG(2n + 1, q)
219(1)
4.7 Notes and references
219(4)
5 Embedded geometries
223(82)
5.1 Polar spaces
223(3)
5.2 Generalised quadrangles
226(6)
5.3 Embedded Shult spaces
232(15)
5.4 Lax and polarised embeddings of Shult spaces
247(6)
5.5 Characterisations of the classical generalised quadrangles
253(13)
5.6 Partial geometries
266(3)
5.7 Embedded partial geometries
269(3)
5.8 (0, α)-geometries and semi-partial geometries
272(9)
5.9 Embedded (0, α)-geometries and semi-partial geometries
281(18)
5.10 Notes and references
299(6)
6 Arcs and caps
305(58)
6.1 Introduction
305(2)
6.2 Caps and codes
307(7)
6.3 The maximum size of a cap for q odd
314(5)
6.4 The maximum size of a cap for q even
319(6)
6.5 General properties of k-arcs and normal rational curves
325(7)
6.6 The maximum size of an arc and the characterisation of such arcs
332(6)
6.7 Arcs and hypersurfaces
338(22)
6.8 Notes and References
360(3)
7 Ovoids, spreads and m-systems of finite classical polar spaces
363(24)
7.1 Finite classical polar spaces
363(1)
7.2 Ovoids and spreads of finite classical polar spaces
364(1)
7.3 Existence of ovoids
365(1)
7.4 Existence of spreads
365(1)
7.5 Open problems
366(1)
7.6 m-systems and partial m-systems of finite classical polar spaces
367(1)
7.7 Intersections with hyperplanes and generators
368(1)
7.8 Bounds on partial m-systems and non-existence of m-systems
369(3)
7.9 m'-systems arising from a given m-system
372(2)
7.10 m-systems, strongly regular graphs and linear projective two-weight codes
374(2)
7.11 m-systems and maximal arcs
376(2)
7.12 Partial m-systems, BLT-sets and sets with the BLT-property
378(2)
7.13 m-systems and SPG-reguli
380(2)
7.14 Small cases
382(1)
7.15 Notes and references
383(4)
References 387(18)
Index 405
Both authors have been active in the field for about a half century. They have authored several books and several hundreds of papers in international journals; they have also been keynote speakers at numerous international conferences, as well as having organised such conferences. The standard works on Galois geometries and finite generalised quadrangles are due to them as authors or co-authors. James Hirschfeld was born and brought up in Sydney, and studied at Sydney and Edinburgh. He has been at Sussex since 1966. Joseph Thas studied at Ghent University, where he has held positions since 1966. He has been a member of the Royal Flemish Academy of Belgium for Science and the Arts since 1988.