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Kummer's Quartic Surface (Cambridge Mathematical Library): R. W. H. Hudson

The theory of surfaces has reached a certain stage of completeness and major efforts concentrate on solving concrete questions rather than further developing the formal theory. Many of these questions are touched on in this classic volume: such as the classification of quartic surfaces, the description of moduli spaces for abelian surfaces, and the automorphism group of a Kummer surface. First printed in 1905 after the untimely death of the author, this work has stood for most of this century as one of the classic reference works in geometry.Table of Contents by culin

Table of Contents

Foreword xi

Prefatory note xxiii

Chapter I. Kummer's Configuration

1.Desmic tetrahedra 1

2.The group of reflexions 4

3.The 16_6 configuration 5

4.The group of sixteen operations 6

5.The incidence diagram 7

6.Linear construction from six arbitrary planes 8

7.Situation of coplanar points 12

Chapter II. The Quartic Surface

8.The Quartic surface with sixteen nodes 14

9.Nomenclature for the nodes and tropes 16

10.The equation of the surface 17

11.The shape of the surface 19

Chapter III. The Orthogonal Matrix Of Linear Forms

12.Preliminary account of matrices 24

13.Orthogonal matrices 26

14.Connection between matrices and quaternions 27

15.The sixteen linear forms 28

16.Quadratic relations 30

17.The ten fundamental quadrics 32

18.The six fundamental complexes 33

19.Irrational equations of Kummer's surface 34

Chapter IV. Line Geometry

20.Polar lines 37

21.Apolar complexes 38

22.Groups of three and four apolar complexes 39

23.Six apolar complexes 40

24.Ten fundamental quadrics 41

25.Klein's 60_15 configuration 42

26.Kummer's 16_6 configuration 44

27.Line coordinates 45

28.Fundamental quadrics 47

29.Fundamental tetrahedra 48

Chapter V. The Quadratic Complex And Congruence

30.Outline of the geometrical theory 50

31.Outline of the algebraical theory 53

32.Elliptic coordinates55 33.Conjugate sets 56

34.Klein's tetrahedra57 35.Relations of lines to 58

36.Asymptotic curves 60

37.Principal asymptotic curves 62

38.The congruence of second order and class 63

39.Singularities of the congruence 63

40.Relation between and [Lambda] 65

41.Confocal congruences 66

Chapter VI. Plucker's Complex Surface

42.Tetrahedral complexes 68

43.Equations of the complex and the complex surface 69

44.Singularities of the surface 71

45.The polar line 72

46.Shape of the surface73

Chapter VII. Sets Of Nodes

47.Group-sets 75

48.Comparison of notations 76

49.Pairs and octads 77

50.Eighty Rosenhain odd tetrads 78

51.Sixty Gopel even tetrads 79

52.Odd and even hexads 80

Chapter VIII. Equations Of Kummer's Surface

53.The equation referred to a fundamental tetrahedron 81

54.The equation referred to a Rosenhain tetrahedron 83

55.Nodal quartic surfaces 86

Chapter IX.Special Forms Of Kummer's Surface

56.The tetrahedroid 89

57.Multiple tetrahedroids 91

58.Battaglini's harmonic complex 94

59.Limiting forms 98

Chapter X.The Wave Surface

60.Definition of the surface 100

61.Apsidal surfaces 101

62.Singularities of the Wave Surface 102

63.Parametric representation 104

64.Tangent planes 106

65.The four parameters 108

66.Curvature 109

67.Asymptotic lines 110

68.Painvin's complex 112

Chapter XI.Reality And Topology

69.Reality of the complexes 115

70.Six real fundamental complexes 118

71.Equations of surfaces I[subscript a], I[subscript b], I[subscript c] 121

72.Four real and two imaginary complexes 122

73.Two real and four imaginary complexes 125

74.Six imaginary complexes 126

Chapter XII.Geometry Of Four Dimensions

75.Linear manifolds 127

76.Construction of the 15_6 configuration from six points in four dimensions 129

77.Analytical methods 130

78.The 16_6 configuration 131

79.General theory of varieties 132

80.Space sections of a certain quartic variety 134

Chapter XIII. Algebraic Curves On The Surface

81.Geometry on a surface 137

82.Algebraic curves on Kummer's surface 138

83.The [Theta]-equation of a curve 141

84.General theorems on curves 142

85.Classification of families of curves 145

86.Linear systems of curves 146

Chapter XIV. Curves Of Different Orders

87.Quartic curves 149

88.Quartics through the same even tetrad 151

89.Quartics through the same odd tetrad 153

90.Sextics through six nodes 154

91.Sextics through ten nodes 157

92.Octavic curves through eight nodes 158

93.Octavic curves through sixteen nodes 159

Chapter XV.Weddle's Surface

94.Birational transformation of surfaces 160

95.Transformation of Kummer's surface 162

96.Quartic surfaces into which Kummer's surface can be transformed 165

97.Weddle's surface 166

98.Equation of Weddle's surface 169

Chapter XVI.Theta Functions

99.Uniformisation of the surface 173

100.Definition of theta functions 175

101.Characteristics and periods 176

102.Identical relations among the double theta functions 179

103.Parametric expression of Kummer's surface 180

104.Theta functions of higher order 182

105.Sketch of the transcendental theory 184

Chapter XVII.Applications Of Abel's Theorem

106.Tangent sections 188

107.Collinear points 190

108.Asymptotic curves 194

109.Inscribed configurations 196

Chapter XVIII.Singular Kummer Surfaces

110.Elliptic surfaces 200

111.Transformation of theta functions 201

112.The invariant 203

113.Parametric curves 204

114.Unicursal curves 206

115.Geomtrical interpretation of the singular relation k[tau subscript 12]=1 208

116.Intermediary functions 210

117.Singular curves 212

118.Singular surfaces with invariant 5 213

119.Singular surfaces with invariant 8 214

120.Birational transformations of Kummer surfaces into themselves 216

Index 221

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