本书是《Classics in Mathematics》系列之一，以现代观点讲述了代数几何知识，将经典代数曲面和现代代数曲面有机结合，很好地表达出了数学的整体性，是同时期很难得的一本代数曲面教材。全书主要内容包括奇点理论和奇点还原；曲线的线性系统；伴随系和不变量理论；算术亏格和Riemann-Roch定理；连续非线性系统；代数曲面的拓扑性质；代数曲面上的单积分和双重积分；复平面上的Branch曲线和连续性。

Chapter 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

Chapter II.Linear Systems of Curves

 t.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

Chapter III.Adjoint Systems and the Theory of Invariants

 1.Complete linear systems of plane curves

 2.Complete linear systems of surfaces in Sa

 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

Chapter 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

Chapter 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 a of SEVERI

 8.On the moduli of algebraic surfaces

Appendix to Chapter V by DAVID MUMFORD

Chapter 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 δi and the invariant cycles

 7.The fundamental homologies for the i-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

Chapter 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 POINCARIE

 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

Chapter VIII.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

Bibliography

Supplementary Bibliography for Second Edition

Index