Author | Zariski, Oscar. author |
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Title | Algebraic Surfaces [electronic resource] / by Oscar Zariski |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1971 |
Edition | Second Supplemented Edition |
Connect to | http://dx.doi.org/10.1007/978-3-642-61991-5 |
Descript | XI, 273 p. online resource |
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 Poincarรฉ -- 5. The existence theorem of Lefschetz-Poincarรฉ -- 6. Reducible integrals. Theorem of Poincarรฉ -- 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