Algebraic surfaces
Author(s)
Bibliographic Information
Algebraic surfaces
（Classics in mathematics）
Springer, c1995
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Note
Reprint of the 1971 ed.
Includes bibliographical references p.[248]-268 and index
Description and Table of Contents
Description
From the reviews: "The author's book [...] saw its first edition in 1935. [...] Now as before, the original text of the book is an excellent source for an interested reader to study the methods of classical algebraic geometry, and to find the great old results. [...] a timelessly beautiful pearl in the cultural heritage of mathematics as a whole." Zentralblatt MATH
Table of Contents
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 Poincare.- 5. The existence theorem of Lefschetz-Poincare.- 6. Reducible integrals. Theorem of Poincare.- 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 Plane Algebraic 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.
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