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Preface; Status of the subject; Related topics; Acknowledgements; Terminology; PART V; 1 Quadrics; 1.1 Canonical forms; 1.2 Invariants; 1.3 Tangency and polarity; 1.4 Generators; 1.5 Numbers of subspaces on a quadric; 1.6 The orthogonal groups; 1.7 The polarity reconsidered; 1.8 Sections of non-singular quadrics; 1.9 Parabolic sections of parabolic quadrics; 1.10 The characterisation of quadrics; 1.11 Further characterisations of quadrics; 1.12 The Principle of Triality; 1.13 Generalised hexagons; 1.14 Notes and references; Sections 1.1-1.9; Section 1.10; Section 1.11
Sections 1.12-1.132 Hermitian varieties; 2.1 Introduction; 2.2 Tangency and polarity; 2.3 Generators and sub-generators; 2.4 Sections of Un; 2.5 The characterisation of Hermitian varieties; 2.6 The characterisation of projections of quadrics; 2.7 Notes and references; Sections 2.1-2.3; Section 2.4; Sections 2.5-2.6; 3 Grassmann varieties; 3.1 Plücker and Grassmann coordinates; 3.2 Grassmann varieties; 3.3 A characterisation of Grassmann varieties; 3.4 Embedding of Grassmann spaces; 3.5 Notes and references; Section 3.1; Section 3.2; Section 3.3; Section 3.4; 4 Veronese and Segre varieties
4.1 Veronese varieties4.2 Characterisations; 4.2.1 Characterisations of V2n n of the first kind; 4.2.2 Characterisations of V2n n of the second kind; 4.2.3 Characterisations of V2n n of the third kind; 4.2.4 Characterisations of V2n n of the fourth kind; 4.3 Hermitian Veroneseans; 4.4 Characterisations of Hermitian Veroneseans; 4.4.1 Characterisations of Hn,n2+2n of the first kind; 4.4.2 Characterisations of Hn,n2+2n of the third kind; 4.4.3 Characterisations of H2,8 of the fourth kind; 4.5 Segre varieties; 4.6 Regular n-spreads and Segre varieties S1; n
4.6.1 Construction method for n-spreads of PG (2n + 1,q)4.7 Notes and references; Section 4.1; Section 4.2; Section 4.3; Section 4.4; Section 4.5; Section 4.6; 5 Embedded geometries; 5.1 Polar spaces; 5.2 Generalised quadrangles; 5.3 Embedded Shult spaces; 5.4 Lax and polarised embeddings of Shult spaces; 5.5 Characterisations of the classical generalised quadrangles; 5.6 Partial geometries; 5.7 Embedded partial geometries; 5.8 (0, α)-geometries and semi-partial geometries; 5.9 Embedded (0, α)-geometries and semi-partial geometries; 5.10 Notes and references; Section 5.1; Section 5.2
Section 5.3Section 5.4; Section 5.5; Section 5.6; Section 5.7; Section 5.8; Section 5.9; 6 Arcs and caps; 6.1 Introduction; 6.2 Caps and codes; 6.3 The maximum size of a cap for q odd; 6.4 The maximum size of a cap for q even; 6.5 General properties of k-arcs and normal rational curves; 6.6 The maximum size of an arc and the characterisation of such arcs; 6.7 Arcs and hypersurfaces; 6.8 Notes and References; Section 6.1; Section 6.2; Section 6.3; Section 6.4; Section 6.5; Section 6.6; Section 6.7; 7 Ovoids, spreads and m-systems of finite classical polar spaces
Sections 1.12-1.132 Hermitian varieties; 2.1 Introduction; 2.2 Tangency and polarity; 2.3 Generators and sub-generators; 2.4 Sections of Un; 2.5 The characterisation of Hermitian varieties; 2.6 The characterisation of projections of quadrics; 2.7 Notes and references; Sections 2.1-2.3; Section 2.4; Sections 2.5-2.6; 3 Grassmann varieties; 3.1 Plücker and Grassmann coordinates; 3.2 Grassmann varieties; 3.3 A characterisation of Grassmann varieties; 3.4 Embedding of Grassmann spaces; 3.5 Notes and references; Section 3.1; Section 3.2; Section 3.3; Section 3.4; 4 Veronese and Segre varieties
4.1 Veronese varieties4.2 Characterisations; 4.2.1 Characterisations of V2n n of the first kind; 4.2.2 Characterisations of V2n n of the second kind; 4.2.3 Characterisations of V2n n of the third kind; 4.2.4 Characterisations of V2n n of the fourth kind; 4.3 Hermitian Veroneseans; 4.4 Characterisations of Hermitian Veroneseans; 4.4.1 Characterisations of Hn,n2+2n of the first kind; 4.4.2 Characterisations of Hn,n2+2n of the third kind; 4.4.3 Characterisations of H2,8 of the fourth kind; 4.5 Segre varieties; 4.6 Regular n-spreads and Segre varieties S1; n
4.6.1 Construction method for n-spreads of PG (2n + 1,q)4.7 Notes and references; Section 4.1; Section 4.2; Section 4.3; Section 4.4; Section 4.5; Section 4.6; 5 Embedded geometries; 5.1 Polar spaces; 5.2 Generalised quadrangles; 5.3 Embedded Shult spaces; 5.4 Lax and polarised embeddings of Shult spaces; 5.5 Characterisations of the classical generalised quadrangles; 5.6 Partial geometries; 5.7 Embedded partial geometries; 5.8 (0, α)-geometries and semi-partial geometries; 5.9 Embedded (0, α)-geometries and semi-partial geometries; 5.10 Notes and references; Section 5.1; Section 5.2
Section 5.3Section 5.4; Section 5.5; Section 5.6; Section 5.7; Section 5.8; Section 5.9; 6 Arcs and caps; 6.1 Introduction; 6.2 Caps and codes; 6.3 The maximum size of a cap for q odd; 6.4 The maximum size of a cap for q even; 6.5 General properties of k-arcs and normal rational curves; 6.6 The maximum size of an arc and the characterisation of such arcs; 6.7 Arcs and hypersurfaces; 6.8 Notes and References; Section 6.1; Section 6.2; Section 6.3; Section 6.4; Section 6.5; Section 6.6; Section 6.7; 7 Ovoids, spreads and m-systems of finite classical polar spaces