Principles of Geometry, Volume 5Henry Frederick Baker (1866-1956) was a renowned British mathematician specialising in algebraic geometry. He was elected a Fellow of the Royal Society in 1898 and appointed the Lowndean Professor of Astronomy and Geometry in the University of Cambridge in 1914. First published between 1922 and 1925, the six-volume Principles of Geometry was a synthesis of Baker's lecture series on geometry and was the first British work on geometry to use axiomatic methods without the use of co-ordinates. The first four volumes describe the projective geometry of space of between two and five dimensions, with the last two volumes reflecting Baker's later research interests in the birational theory of surfaces. The work as a whole provides a detailed insight into the geometry which was developing at the time of publication. This, the fifth volume, describes the birational geometry of curves. |
Contents
INTRODUCTORY ACCOUNT | 1 |
Greatest possible number of double points of a plane curve 1011 | 10 |
Examples of elliptic curves Coresiduation Salmons | 18 |
THE ELIMINATION OF THE MULTIPLE | 24 |
Examples of transformation of curves 3134 | 31 |
The parametric expression of a branch of a curve 3943 | 39 |
General theorem for infinities of a rational function 4649 | 46 |
Examples of Abels theorem 5557 | 55 |
Riemann surfaces 121124 | 121 |
Curve systems on a surface The EulerPoincaré invariant 132133 | 132 |
INTEGRALS RELATIONS AMONG PERIODS | 136 |
THE MODULAR EXPRESSION | 147 |
The functions reciprocal to the fundamental integral func | 157 |
The structure of a certain fundamental rational function | 169 |
ENUMERATIVE PROPERTIES | 182 |
Curves which are the complete intersection of two surfaces | 201 |
MENTALS OF THE THEORY OF LINEAR SERIES | 59 |
Equivalent or coresidual sets of points on the curve | 65 |
Applications of the RiemannRoch formula Cliffords | 78 |
The existence of a rational function with assigned poles | 86 |
The theory of special sets an extension of Cliffords theorem | 94 |
Related theorem as to intersections of plane curves 105107 | 105 |
THE PERIODS OF ALGEBRAIC INTEGRALS | 111 |
Curves which are the partial intersection of two surfaces | 208 |
Examples Curves not determinable by three surfaces | 209 |
Examples Composite curve intersection of two surfaces | 216 |
Another proof of the determination of the canonical series | 226 |
The greatest genus possible for a curve of given order | 234 |
243 | |
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Common terms and phrases
a₁ Abel's Theorem adjoint curve aggregate algebraic curve algebraic integral arbitrary points b₁ birational boundary curve branch place branch points C₂ canonical curve canonical series circuit closed curve coefficients complete intersection conic consider constants contain coordinates coresidual corresponding cubic curve cubic surface curve C₁ curve f curve in space curve of order cusp denote determined dimensions double points equation everywhere finite integrals expression formula freedom fundamental curve g₁ genus given curve homogeneous infinities inflexions linear series linearly independent meet the curve multiple points n₁ neighbourhood number of points obtained order n-3 original curve osculating plane oval cut p-curves parameter pass plane curve points of F=0 poles polynomial power series proved quadric surface quartic quintic curve rational function Riemann surface ruled surface series of sets sextic curve shewn suppose surfaces of order tangent line theorem vanishes variable y₁ zero