Algebraic Surfaces

Front Cover
Springer Science & Business Media, Feb 15, 1995 - Mathematics - 270 pages
The aim of the present monograph is to give a systematic exposition of the theory of algebraic surfaces emphasizing the interrelations between the various aspects of the theory: algebro-geometric, topological and transcendental. To achieve this aim, and still remain inside the limits of the allotted space, it was necessary to confine the exposition to topics which are absolutely fundamental. The present work therefore makes no claim to completeness, but it does, however, cover most of the central points of the theory. A presentation of the theory of surfaces, to be effective at all, must above all give the typical methods of proof used in the theory and their underlying ideas. It is especially true of algebraic geometry that in this domain the methods employed are at least as important as the results. The author has therefore avoided, as much as possible, purely formal accounts of results. The proofs given are of necessity condensed, for reasons of space, but no attempt has been made to condense them beyond the point of intelligibility. In many instances, due to exigencies of simplicity and rigor, the proofs given in the text differ, to a greater or less extent, from the proofs given in the original papers.
 

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Contents

Theory and Reduction of Singularities
viii
3 Singularities of space algebraic curves
5
4 Topological classification of singularities
7
5 Singularities of algebraic surfaces
8
6 The reduction of singularities of an algebraic surface
12
Linear Systems of Curves
19
2 On the conditions imposed by infinitely near base points
22
3 Complete linear systems
24
7 The fundamental homologies for the 1cycles on F
130
8 The reduction of F to a cell
132
9 The threedimensional cycles
133
10 The twodimensional cycles Every chain C2 on L
134
11 The group of torsion
135
12 Homologies between algebraic cycles and algebraic equivalence The invariant 0
137
13 The topological theory of algebraic correspondences
138
Appendix to Chapter VI
142

4 Addition and subtraction of linear systems
26
5 The virtual characters of an arbitrary linear system
29
6 Exceptional curves
31
7 Invariance of the virtual characters
36
8 Virtual characteristic series Virtual curves
38
Appendix to Chapter II
40
Adjoint Systems and the Theory of Invariants
46
3 Subadjoint surfaces
48
4 Subadjoint systems of a given linear system
50
5 The distributive property of subadjunction
53
6 Adjoint systems
55
7 The residue theorem in its projective form
61
9 The pluricanonical systems
65
Appendix to Chapter III
66
The Arithmetic Genus and the Generalized Theorem of RIEMANNROCH
70
2 The theorem of RlEMANNROCH on algebraic surfaces
72
3 The deficiency of the characteristic series of a complete linear system
75
4 The elimination of exceptional curves and the characterization of ruled surfaces
78
Appendix to Chapter IV
83
Continuous Nonlinear Systems
87
2 Complete continuous systems and algebraic equivalence
90
3 The completeness of the characteristic series of a complete continuous system
93
4 The variety of PICARD
99
6 The theory of the base and the number ϱ of PICARO
102
7 The division group and the invariant σ of SEVERI
106
8 On the moduli of algebraic surfaces
108
Appendix to Chapter V
113
Topological Properties of Algebraic Surfaces
124
3 Algebraic cycles on F and their intersections
125
4 The representation of F upon a multiple plane
126
5 The deformation of a variable plane section of F
127
6 The vanishing cycles and the invariant cycles
128
Simple and Double Integrals on an Algebraic Surface
151
2 Simple integrals of the second kind
152
3 On the number of independent simple integrals of the first and of the second kind attached to a surface of irregularity q The fundamental theorem
154
4 The normal functions of POINCAR…
160
5 The existence theorem of LEFSCHETZPOINCAR…
164
6 Reducible integrals Theorem of POINCAR…
168
7 Miscellaneous applications of the existence theorem
172
8 Double integrals of the first kind Theorem of HODGE
177
9 Residues of double integrals and the reduction of the double integrals of the second kind
181
10 Normal double integrals and the determination of the number of independent double integrals of the second kind
186
Appendix to Chapter VII
192
Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves
202
2 Properties of the fundamental group G
205
3 The irregularity of cyclic multiple planes
206
4 Complete continuous systems of plane curves with d nodes
209
5 Continuous systems of plane algebraic curves with nodes and cusps
214
Appendix 1 to Chapter VIII
219
Appendix 2 to Chapter VIII
224
Series of Equivalence
227
2 Series of equivalence
228
3 Invariant series of equivalence
230
4 Topological and transcendental properties of series of equivalence
232
5 Added in 2nd edition by D Mumford
233
Correspondences between Algebraic Varieties
234
2 The transcendental equations and the rank of a correspondence
236
3 The case of two coincident varieties Correspondences with valence
239
4 The principle of correspondence of ZEUTHBNSEVERI
240
Bibliography
243
Supplementary Bibliography for Second Edition
251
Index
264
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About the author (1995)

Biography of Oscar Zariski

Oscar Zariski (24.4.1899-4.7.1986) was born in Kobryn, Poland, and studied at the universities of Kiev and Rome. He held positions at Rome University, John Hopkins University, the University of Illinois and from 1947 at Harvard University.

Zariski's main fields of activity were in algebraic geometry, algebra, algebraic function theory and topology. His most influential results bear on algebraic surfaces, the resolution of singularities and the foundations of algebraic geometry over arbitrary fields.

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