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Chapter Ⅰ．Theory and Reduction of Singularities1

1．Algebraic varieties and birational transformations1

2．Singularities of plane algebraic curves5

3．Singularities of space algebraic curves10

4．Topological classification of singularities12

5．Singularities of algebraic surfaces13

6．The reduction of singularities of an algebraic surface17

Chapter Ⅱ．Linear Systems of Curves24

1．Definitions and general properties24

2．On the conditions imposed by infinitely near base points27

3．Complete linear systems29

4．Addition and subtraction of linear systems31

5．The virtual characters of an arbitrary linear system34

6．Exceptional curves36

7．Invariance of the virtual characters41

8．Virtual characteristic series.Virtual curves43

Appendix to Chapter Ⅱ by JOSEPH LIPMAN45

Chapter Ⅲ．Adjoint Systems and the Theory of Invariants51

1．Complete linear systems of plane curves51

2．Complete linear systems of surfaces in S351

3．Subadjoint surfaces53

4．Subadjoint systems of a given linear system55

5．The distributive property of subadjunction58

6．Adjoint systems60

7．The residue theorem in its projective form66

8．The canonical system66

9．The pluricanonical systems70

Appendix to Chapter Ⅲ by DAVID MUMFORD71

Chapter Ⅳ．The Arithmetic Genus and the Generalized Theorem of RIEMANN-ROCH75

1．The arithmetic genus pa75

2．The theorem of RIEMANN-ROCH on algebraic surfaces77

3．The deficiency of the characteristic series of a complete linear system80

4．The elimination of exceptional curves and the characterization of ruled surfaces83

Appendix to Chapter Ⅳ by DAVID MUMFORD88

Chapter Ⅴ．Continuous Non-linear Systems92

1．Definitions and general properties92

2．Complete continuous systems and algebraic equivalence95

3．The completeness of the characteristic series of a complete continuous system98

4．The variety of PICARD104

5．Equivalence criteria104

6．The theory of the base and the number ？ of PICARD107

7．The division group and the invariant σ of SEVERI111

8．On the moduli of algebraic surfaces113

Appendix to Chapter Ⅴ by DAVID MUMFORD118

Chapter Ⅵ．Topological Properties of Algebraic Surfaces129

1．Terminology and notations129

2．An algebraic surface as a manifold M4129

3．Algebraic cycles on F and their intersections130

4．The representation of F upon a multiple plane131

5．The deformation of a variable plane section of F132

6．The vanishing cycles δi and the invariant cycles133

7．The fundamental homologies for the 1-cycles on F135

8．The reduction of F to a cell137

9．The three-dimensional cycles138

10．The two-dimensional cycles139

11．The group of torsion140

12．Homologies between algebraic cycles and algebraic equivalence.The invariant ？0142

13．The topological theory of algebraic correspondences143

Appendix to Chapter Ⅵ by DAVID MUMFORD147

Chapter Ⅶ．Simple and Double Integrals on an Algebraic Surface156

1．Classification of integrals156

2．Simple integrals of the second kind157

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 theorem159

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 theorem177

8．Double integrals of the first kind.Theorem of HODGE182

9．Residues of double integrals and the reduction of the double integrals of the second kind186

10．Normal double integrals and the determination of the number of independent double integrals of the second kind191

Appendix to Chapter Ⅶ by DAVID MUMFORD197

Chapter Ⅷ．Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves207

1．The problem of existence of algebraic functions of two variables207

2．Properties of the fundamental group G210

3．The irregularity of cyclic multiple planes211

4．Complete continuous systems of plane curves with d nodes214

5．Continuous systems of plane algebraic curves with nodes and cusps219

Appendix 1 to Chapter Ⅷ by SHREERAM SHANKAR ABHYANKAR224

Appendix 2 to Chapter Ⅷ by DAVID MUMFORD229

Appendix A．Series of Equivalence232

1．Equivalence between sets of points232

2．Series of equivalence233

3．Invariant series of equivalence235

4．Topological and transcendental properties of series of equivalence237

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

Appendix B．Correspondences between Algebraic Varieties239

1．The fixed point formula of LEFSCHETZ239

2．The transcendental equations and the rank of a correspondence241

3．The case of two coincident varieties.Correspondences with valence244

4．The principle of correspondence of ZEUTHEN-SEVERI245

Bibliography248

Supplementary Bibliography for Second Edition256

Index269