Kummer's Quartic Surface

Front Cover
Cambridge University Press, Oct 11, 1990 - Mathematics - 222 pages
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.
 

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Contents

CHAPTER I
1
SECT PAOB 1 Desmic tetrahedra
3
The group of reflexions
4
The 166 configuration
5
The group of sixteen operations
6
The incidence diagram
7
Linear construction from six arbitrary planes
8
Situation of coplanar points
12
Parametric representation
104
Tangent planes
106
The four parameters
108
Curvature
109
Asymptotic lines
111
Painvins complex
112
CHAPTER XI
115
Six real fundamental complexes
118

CHAPTER II
14
Nomenclature for the nodes anil tropes
16
The equation of the surface
18
The shape of the surface
19
CHAPTER III
24
Orthogonal matrices
26
Connection between matrices and quaternions
27
SECT PAQE 15 The sixteen linear forms
28
Quadratic relations
30
The ten fundamental quadrics
32
The six fundamental complexes
33
Irrational equations of Rummers surface
34
CHAPTER IV
37
Apolar complexes
38
Groups of three and four apolar complexes
39
Six apolar complexes
40
Ten fundamental quadrics
41
Kleins 6016 configuration
42
Rummers 16B configuration
44
Line coordinates
45
Fundamental quadrios
47
Fundamental tetrahedra
49
CHAPTER V
51
Outline of the algebraical theory
53
Elliptic coordinates
55
Conjugate sets
56
Rleins tetrahedra
57
Relations of lines to
58
Asymptotic curves
60
Principal asymptotic curves
62
The congruence of second order and class
63
Relation between and A
65
Confocal congruences
66
CHAPTER VI
68
Equations of the complex and the complex surface
69
Singularities of the surface
71
The polar line
72
Shape of the surface
73
SECT PAGE 47 Groupsets 75
77
Eighty Rosenhain odd tetrads
78
Sixty Gopel even tetrads
79
Odd and even hexads
80
CHAPTER VIII
81
The equation referred to a Rosenhain tetrahedron
83
Nodal quartio surfaces
86
SPECIAL FORMS OF KUMMERs SURFACE 56 The tetrahedroid
89
Multiple tetrahedroids
93
Battaglinis harmonic complex
97
Limiting forms
98
CHAPTER X
100
Apsidal surfaces
101
Singularities of the Wave Surface
102
Equations of surfaces Jn fb Ic
121
Four real and two imaginary complexes
122
Two real and four imaginary complexes
125
Six imaginary complexes
126
CHAPTER XII
127
Construction of the 150 configuration from six points in four dimensions
129
Analytical methods
130
The 166 configuration
131
General theory of varieties
132
Space sections of a certain quartic variety
134
CHAPTER XIII
137
Algebraic curves on Rummers surface
138
The 8equation of a curve
141
General theorems on curves
142
Classification of families of curves
145
Linear systems of curves
146
CHAPTER XIV
149
Quartics through the same even tetrad
151
Quartics through the same odd tetrad
153
Sextics through six nodes
154
Sextics through ten nodes
157
Octavic curves through eight nodes
158
Octavic curves through sixteen nodes
159
CHAPTER XV
160
Transformation of Rummers surface
163
Quartio surfaces into which Rummers surface can be trans formed
165
Weddles surface
166
Equation of Weddles surface
169
BKCT PAGE 99 Uniformisation of the surface
173
Definition of theta functions
175
Characteristics and periods
176
Identical relations among the double theta functions
179
Parametric expression of Rummers surface
180
Theta functions of higher order
182
Sketch of the transcendental theory
184
CHAPTER XVII
188
Collinear points
190
Asymptotic curves
194
Inscribed configurations
196
CHAPTER XVIII
200
Transformation of theta functions
201
The invariant
203
Parametric curves
204
Unicursal curves
206
Geometrical interpretation of the singular relation tri2l
208
Intermediary functions
210
Singular curves
212
Singular surfaces with invariant 6
213
Singular surfaces with invariant 8
214
Birational transformations of Rummer surfaces into themselves
216
INDEX
221
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