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

Kummer's Quartic Surface

Cambridge Mathematical Library

By: R. W. H. Hudson, W. Barth (Foreword by), H. F. Baker (Preface by)

Paperback Published: 10th December 1990
ISBN: 9780521397902
Number Of Pages: 252

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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.

"This famous book is a prototype for the possibility of explaining and exploring a many-faceted topic of research, without focussing on general definitions, formal techniques, or even fancy machinery. In this regard, the book still stands as a highly recommendable, unparalleled introduction to Kummer surfaces, as a permanent source of inspiration and, last but not least. as an everlasting symbol of mathematical culture." Werner Kleinert, Mathematical Reviews

Forewordp. xi
Prefatory notep. xxiii
Kummer's Configuration
Desmic tetrahedrap. 1
The group of reflexionsp. 4
The 16[subscript 6] configurationp. 5
The group of sixteen operationsp. 6
The incidence diagramp. 7
Linear construction from six arbitrary planesp. 8
Situation of coplanar pointsp. 12
The Quartic Surface
The Quartic surface with sixteen nodesp. 14
Nomenclature for the nodes and tropesp. 16
The equation of the surfacep. 17
The shape of the surfacep. 19
The Orthogonal Matrix Of Linear Forms
Preliminary account of matricesp. 24
Orthogonal matricesp. 26
Connection between matrices and quaternionsp. 27
The sixteen linear formsp. 28
Quadratic relationsp. 30
The ten fundamental quadricsp. 32
The six fundamental complexesp. 33
Irrational equations of Kummer's surfacep. 34
Line Geometry
Polar linesp. 37
Apolar complexesp. 38
Groups of three and four apolar complexesp. 39
Six apolar complexesp. 40
Ten fundamental quadricsp. 41
Klein's 60[subscript 15] configurationp. 42
Kummer's 16[subscript 6] configurationp. 44
Line coordinatesp. 45
Fundamental quadricsp. 47
Fundamental tetrahedrap. 48
The Quadratic Complex And Congruence
Outline of the geometrical theoryp. 50
Outline of the algebraical theoryp. 53
Elliptic coordinatesp. 55
Conjugate setsp. 56
Klein's tetrahedrap. 57
Relations of lines to [Phi]p. 58
Asymptotic curvesp. 60
Principal asymptotic curvesp. 62
The congruence of second order and classp. 63
Singularities of the congruencep. 63
Relation between [Phi] and [Lambda]p. 65
Confocal congruencesp. 66
Plucker's Complex Surface
Tetrahedral complexesp. 68
Equations of the complex and the complex surfacep. 69
Singularities of the surfacep. 71
The polar linep. 72
Shape of the surfacep. 73
Sets Of Nodes
Group-setsp. 75
Comparison of notationsp. 76
Pairs and octadsp. 77
Eighty Rosenhain odd tetradsp. 78
Sixty Gopel even tetradsp. 79
Odd and even hexadsp. 80
Equations Of Kummer's Surface
The equation referred to a fundamental tetrahedronp. 81
The equation referred to a Rosenhain tetrahedronp. 83
Nodal quartic surfacesp. 86
Special Forms Of Kummer's Surface
The tetrahedroidp. 89
Multiple tetrahedroidsp. 91
Battaglini's harmonic complexp. 94
Limiting formsp. 98
The Wave Surface
Definition of the surfacep. 100
Apsidal surfacesp. 101
Singularities of the Wave Surfacep. 102
Parametric representationp. 104
Tangent planesp. 106
The four parametersp. 108
Curvaturep. 109
Asymptotic linesp. 110
Painvin's complexp. 112
Reality And Topology
Reality of the complexesp. 115
Six real fundamental complexesp. 118
Equations of surfaces I[subscript a], I[subscript b], I[subscript c]p. 121
Four real and two imaginary complexesp. 122
Two real and four imaginary complexesp. 125
Six imaginary complexesp. 126
Geometry Of Four Dimensions
Linear manifoldsp. 127
Construction of the 15[subscript 6] configuration from six points in four dimensionsp. 129
Analytical methodsp. 130
The 16[subscript 6] configurationp. 131
General theory of varietiesp. 132
Space sections of a certain quartic varietyp. 134
Algebraic Curves On The Surface
Geometry on a surfacep. 137
Algebraic curves on Kummer's surfacep. 138
The [Theta]-equation of a curvep. 141
General theorems on curvesp. 142
Classification of families of curvesp. 145
Linear systems of curvesp. 146
Curves Of Different Orders
Quartic curvesp. 149
Quartics through the same even tetradp. 151
Quartics through the same odd tetradp. 153
Sextics through six nodesp. 154
Sextics through ten nodesp. 157
Octavic curves through eight nodesp. 158
Octavic curves through sixteen nodesp. 159
Weddle's Surface
Birational transformation of surfacesp. 160
Transformation of Kummer's surfacep. 162
Quartic surfaces into which Kummer's surface can be transformedp. 165
Weddle's surfacep. 166
Equation of Weddle's surfacep. 169
Theta Functions
Uniformisation of the surfacep. 173
Definition of theta functionsp. 175
Characteristics and periodsp. 176
Identical relations among the double theta functionsp. 179
Parametric expression of Kummer's surfacep. 180
Theta functions of higher orderp. 182
Sketch of the transcendental theoryp. 184
Applications Of Abel's Theorem
Tangent sectionsp. 188
Collinear pointsp. 190
Asymptotic curvesp. 194
Inscribed configurationsp. 196
Singular Kummer Surfaces
Elliptic surfacesp. 200
Transformation of theta functionsp. 201
The invariantp. 203
Parametric curvesp. 204
Unicursal curvesp. 206
Geomtrical interpretation of the singular relation k[tau subscript 12]=1p. 208
Intermediary functionsp. 210
Singular curvesp. 212
Singular surfaces with invariant 5p. 213
Singular surfaces with invariant 8p. 214
Birational transformations of Kummer surfaces into themselvesp. 216
Indexp. 221
Plate
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521397902
ISBN-10: 0521397901
Series: Cambridge Mathematical Library
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 252
Published: 10th December 1990
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.5
Weight (kg): 0.38

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