An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this book explores fundamental concepts of the general theory of algebraic varieties: general point, dimension, function field, rational transformations, and correspondences as well as formal power series and an extensive survey of algebraic curves. 1953 edition.

Preface

v

Algebraic Foundations

Preliminaries

3

(2)

Resultants and elimination

5

(1)

Algebraic dependence. Transcendency

6

(1)

Extension of the groundfield

7

(3)

Differentials (characteristic zero)

10

(6)

Algebraic Varieties: Fundamental Concepts

Affine and projective spaces

16

(5)

Algebraic varieties and their ideals

21

(5)

General points. Dimension. Function field

26

(7)

Projections

33

(2)

Differentials. Singular points. Tangent spaces (ground-field of characteristic zero)

35

(5)

Some intersection properties

40

(7)

Transformations of Algebraic Varieties

Rational transformations

47

(2)

Birational transformations

49

(4)

Normal systems

53

(8)

Product spaces

61

(3)

Algebraic correspondences

64

(8)

Complements on intersections

72

(2)

Appendix: Groundfield of characteristic p ≠ 0

74

(4)

Formal Power Series

Basic concepts and theorems

78

(3)

Algebroid varieties

81

(8)

Local properties of algebraic varieties

89

(7)

Algebraic varieties as topological spaces

96

(2)

Algebraic Curves, their Places and Transformations

Formal power series in one and two variables

98

(1)

Puiseux's theorem

99

(5)

The places of an algebraic curve

104

(4)

Valuations

108

(4)

Multiple points, intersections and the places

112

(5)

Rational and birational transformations and the places

117

(4)

Space curves

121

(6)

Reduction of singularities

127

(8)

Linear Series

Divisors and their classes

135

(2)

Linear series: First properties

137

(5)

Birational models and linear series

142

(2)

Rational, elliptic and hyperelliptic curves

144

(1)

Adjoint curves and series

145

(4)

The theorem of Riemann-Roch

149

(6)

Abelian Differentials

Preliminary questions

155

(3)

The divisors of the differentials. Differentials of the first kind

158

(2)

Elliptic and hyperelliptic differentials. Canonical model

160

(5)

Differentials of the second and third kinds

165

(7)

Jacobian series

172

(4)

Abel's Theorem. Algebraic Series and Correspondences

Abel's theorem

176

(2)

Algebraic series

178

(3)

Algebraic correspondences between two curves

181

(8)

Algebraic correspondences of a curve with itself

189

(5)

Products of correspondences

194

(2)

Systems of Curves on a Surface

Generalities on the curves on a surface

196

(5)

Differentials of the surface Φ

201

(2)

Simple differentials

203

(2)

Double differentials

205

(11)

Algebraic dependence of curves on a surface according to Severi

216

(4)

Surface product of two curves. Application to correspondences

220

(2)

Birational invariance

222

(2)

Appendix

224

(2)

Bibliography

226

(3)

List of symbols most frequently used in the text

229

(2)

Index

231

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