内容

目次

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 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.