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Chapter Ⅰ．Theory and Reduction of Singularities 1

1．Algebraic varieties and birational transformations 1

2．Singularities of plane algebraic curves 5

3．Singularities of space algebraic curves 10

4．Topological classification of singularities 12

5．Singularities of algebraic surfaces 13

6．The reduction of singularities of an algebraic surface 17

Chapter Ⅱ．Linear Systems of Curves 24

1．Definitions and general properties 24

2．On the conditions imposed by infinitely near base points 27

3．Complete linear systems 29

4．Addition and subtraction of linear systems 31

5．The virtual characters of an arbitrary linear system 34

6．Exceptional curves 36

7．Invariance of the virtual characters 41

8．Virtual characteristic series.Virtual curves 43

Appendix to Chapter Ⅱ by JOSEPH LIPMAN 45

Chapter Ⅲ．Adjoint Systems and the Theory of Invariants 51

1．Complete linear systems of plane curves 51

2．Complete linear systems of surfaces in S3 51

3．Subadjoint surfaces 53

4．Subadjoint systems of a given linear system 55

5．The distributive property of subadjunction 58

6．Adjoint systems 60

7．The residue theorem in its projective form 66

8．The canonical system 66

9．The pluricanonical systems 70

Appendix to Chapter Ⅲ by DAVID MUMFORD 71

Chapter Ⅳ．The Arithmetic Genus and the Generalized Theorem of RIEMANN-ROCH 75

1．The arithmetic genus pa 75

2．The theorem of RIEMANN-ROCH on algebraic surfaces 77

3．The deficiency of the characteristic series of a complete linear system 80

4．The elimination of exceptional curves and the characterization of ruled surfaces 83

Appendix to Chapter Ⅳ by DAVID MUMFORD 88

Chapter Ⅴ．Continuous Non-linear Systems 92

1．Definitions and general properties 92

2．Complete continuous systems and algebraic equivalence 95

3．The completeness of the characteristic series of a complete continuous system 98

4．The variety of PICARD 104

5．Equivalence criteria 104

6．The theory of the base and the number ？ of PICARD 107

7．The division group and the invariant σ of SEVERI 111

8．On the moduli of algebraic surfaces 113

Appendix to Chapter Ⅴ by DAVID MUMFORD 118

Chapter Ⅵ．Topological Properties of Algebraic Surfaces 129

1．Terminology and notations 129

2．An algebraic surface as a manifold M4 129

3．Algebraic cycles on F and their intersections 130

4．The representation of F upon a multiple plane 131

5．The deformation of a variable plane section of F 132

6．The vanishing cycles δi and the invariant cycles 133

7．The fundamental homologies for the 1-cycles on F 135

8．The reduction of F to a cell 137

9．The three-dimensional cycles 138

10．The two-dimensional cycles 139

11．The group of torsion 140

12．Homologies between algebraic cycles and algebraic equivalence.The invariant ？0 142

13．The topological theory of algebraic correspondences 143

Appendix to Chapter Ⅵ by DAVID MUMFORD 147

Chapter Ⅶ．Simple and Double Integrals on an Algebraic Surface 156

1．Classification of integrals 156

2．Simple integrals of the second kind 157

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 159

4．The normal functions of POINCAR？ 165

5．The existence theorem of LEFSCHETZ-POINCAR？ 169

6．Reducible integrals.Theorem of POINCAR？ 173

7．Miscellaneous applications of the existence theorem 177

8．Double integrals of the first kind.Theorem of HODGE 182

9．Residues of double integrals and the reduction of the double integrals of the second kind 186

10．Normal double integrals and the determination of the number of independent double integrals of the second kind 191

Appendix to Chapter Ⅶ by DAVID MUMFORD 197

Chapter Ⅷ．Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves 207

1．The problem of existence of algebraic functions of two variables 207

2．Properties of the fundamental group G 210

3．The irregularity of cyclic multiple planes 211

4．Complete continuous systems of plane curves with d nodes 214

5．Continuous systems of plane algebraic curves with nodes and cusps 219

Appendix 1 to Chapter Ⅷ by SHREERAM SHANKAR ABHYANKAR 224

Appendix 2 to Chapter Ⅷ by DAVID MUMFORD 229

Appendix A．Series of Equivalence 232

1．Equivalence between sets of points 232

2．Series of equivalence 233

3．Invariant series of equivalence 235

4．Topological and transcendental properties of series of equivalence 237

5．(Added in 2nd edition,by D.MUMFORD) 238

Appendix B．Correspondences between Algebraic Varieties 239

1．The fixed point formula of LEFSCHETZ 239

2．The transcendental equations and the rank of a correspondence 241

3．The case of two coincident varieties.Correspondences with valence 244

4．The principle of correspondence of ZEUTHEN-SEVERI 245

Bibliography 248

Supplementary Bibliography for Second Edition 256

Index 269