## Kummer's Quartic SurfaceThe 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

221 | |

### Other editions - View all

### Common terms and phrases

abelian algebraic algebraic curve apolar complexes asymptotic curve base points belong bitangent coefficients coincide collinear columns complete intersection complex cone configuration congruence contains coplanar corners corresponding cosingular cross ratio cubic surface deduced denote desmic system determined directrices edges elements elliptic coordinates equation expressed focal surface four nodes four points fundamental quadrics geometry given group-set Hence hyperbolic segments identity incidence diagram inflexional tangents inscribed Kummer surface linear complexes linear forms linearly Math nodal line null-plane null-point obtained octad operations orthogonal matrix pairs parameters pass permutable point of contact points and planes polar lines quadratic complex quartic surface rational relation Rummer's sextic singular conic singular planes singular points singular ray singular surface six nodes six planes six points sixteen nodes surface of order symbols tangent plane tangent space tetrad tetrahedron theorem theta functions touch tropes Wave surface