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.

Autorentext

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.

Inhalt
I. Theory and Reduction of Singularities.- 1. Algebraic varieties and birational transformations.- 2. Singularities of plane algebraic curves.- 3. Singularities of space algebraic curves.- 4. Topological classification of singularities.- 5. Singularities of algebraic surfaces.- 6. The reduction of singularities of an algebraic surface.- II. Linear Systems of Curves.- 1. Definitions and general properties.- 2. On the conditions imposed by infinitely near base points.- 3. Complete linear systems.- 4. Addition and subtraction of linear systems.- 5. The virtual characters of an arbitrary linear system.- 6. Exceptional curves.- 7. Invariance of the virtual characters.- 8. Virtual characteristic series. Virtual curves.- Appendix to Chapter II by Joseph Lipman.- III. Adjoint Systems and the Theory of Invariants.- 1. Complete linear systems of plane curves.- 2. Complete linear systems of surfaces in S3.- 3. Subadjoint surfaces.- 4. Subadjoint systems of a given linear system.- 5. The distributive property of subadjunction.- 6. Adjoint systems.- 7. The residue theorem in its projective form.- 8. The canonical system.- 9. The pluricanonical systems.- Appendix to Chapter III by David Mumford.- IV. The Arithmetic Genus and the Generalized Theorem of Riemann-Roch.- 1. The arithmetic genus Pa.- 2. The theorem of Riemann-Roch on algebraic surfaces.- 3. The deficiency of the characteristic series of a complete linear system.- 4. The elimination of exceptional curves and the characterization of ruled surfaces.- Appendix to Chapter IV by David Mumford.- V. Continuous Non-linear Systems.- 1. Definitions and general properties.- 2. Complete continuous systems and algebraic equivalence.- 3. The completeness of the characteristic series of a complete continuous system.- 4. The variety of Picard.- 5. Equivalence criteria.- 6. The theory of the base and the number ? of Picard.- 7. The division group and the invariant ? of Severi.- 8. On the moduli of algebraic surfaces.- Appendix to Chapter V by David Mumford.- VI. Topological Properties of Algebraic Surfaces.- 1. Terminology and notations.- 2. An algebraic surface as a manifold M4.- 3. Algebraic cycles on F and their intersections.- 4. The representation of F upon a multiple plane.- 5. The deformation of a variable plane section of F.- 6. The vanishing cycles ?i, and the invariant cycles.- 7. The fundamental homologies for the 1-cycles on F.- 8. The reduction of F to a cell.- 9. The three-dimensional cycles.- 10. The two-dimensional cycles.- 11. The group of torsion.- 12. Homologies between algebraic cycles and algebraic equivalence. The invariant ?0.- 13. The topological theory of algebraic correspondences.- Appendix to Chapter VI by David Mumford.- VII. Simple and Double Integrals on an Algebraic Surface.- 1. Classification of integrals.- 2. Simple integrals of the second kind.- 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.- 4. The normal functions of Poincaré.- 5. The existence theorem of Lefschetz-Poincaré.- 6. Reducible integrals. Theorem of Poincaré.- 7. Miscellaneous applications of the existence theorem.- 8. Double integrals of the first kind. Theorem of Hodge.- 9. Residues of double integrals and the reduction of the double integrals of the second kind.- 10. Normal double integrals and the determination of the number of independent double integrals of the second kind.- Appendix to Chapter VII by David Mumford.- ChapterVIII. Branch Curves of Multiple Planes and Continuous Systems of PlaneAlgebraic Curves.- 1. The problem of existence of algebraic functions of two variables.- 2. Properties of the fundamental group G.- 3. The irregularity of cyclic multiple planes.- 4. Complete continuous systems of plane curves with d nodes.- 5. Continuous systems of plane algebraic curves with nodes and cusps.- Appendix 1 to Chapter VIII by Shreeram Shankar Abhyankar.- Appendix 2 to Chapter VIII by David Mumford.- Appendix A. Series of Equivalence.- 1. Equivalence between sets of points.- 2. Series of equivalence.- 3. Invariant series of equivalence.- 4. Topological and transcendental properties of series of equivalence.- 5. (Added in 2nd edition, by D. Mumford).- Appendix B. Correspondences between Algebraic Varieties.- 1. The fixed point formula of Lefschetz.- 2. The transcendental equations and the rank of a correspondence.- 3. The case of two coincident varieties. Correspondences with valence.- 4. The principle of correspondence of Zeuthen-Severi.- Supplementary Bibliography for Second Edition.