## Algebraic SurfacesThe aim of the present monograph is to give a systematic exposition of the theory of algebraic surfaces emphasizing the interrelations between the various aspects of the theory: algebro-geometric, topological and transcendental. To achieve this aim, and still remain inside the limits of the allotted space, it was necessary to confine the exposition to topics which are absolutely fundamental. The present work therefore makes no claim to completeness, but it does, however, cover most of the central points of the theory. A presentation of the theory of surfaces, to be effective at all, must above all give the typical methods of proof used in the theory and their underlying ideas. It is especially true of algebraic geometry that in this domain the methods employed are at least as important as the results. The author has therefore avoided, as much as possible, purely formal accounts of results. The proofs given are of necessity condensed, for reasons of space, but no attempt has been made to condense them beyond the point of intelligibility. In many instances, due to exigencies of simplicity and rigor, the proofs given in the text differ, to a greater or less extent, from the proofs given in the original papers. |

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### Contents

Theory and Reduction of Singularities | viii |

3 Singularities of space algebraic curves | 5 |

4 Topological classification of singularities | 7 |

5 Singularities of algebraic surfaces | 8 |

6 The reduction of singularities of an algebraic surface | 12 |

Linear Systems of Curves | 19 |

2 On the conditions imposed by infinitely near base points | 22 |

3 Complete linear systems | 24 |

7 The fundamental homologies for the 1cycles on F | 130 |

8 The reduction of F to a cell | 132 |

9 The threedimensional cycles | 133 |

10 The twodimensional cycles Every chain C2 on L | 134 |

11 The group of torsion | 135 |

12 Homologies between algebraic cycles and algebraic equivalence The invariant 0 | 137 |

13 The topological theory of algebraic correspondences | 138 |

Appendix to Chapter VI | 142 |

4 Addition and subtraction of linear systems | 26 |

5 The virtual characters of an arbitrary linear system | 29 |

6 Exceptional curves | 31 |

7 Invariance of the virtual characters | 36 |

8 Virtual characteristic series Virtual curves | 38 |

Appendix to Chapter II | 40 |

Adjoint Systems and the Theory of Invariants | 46 |

3 Subadjoint surfaces | 48 |

4 Subadjoint systems of a given linear system | 50 |

5 The distributive property of subadjunction | 53 |

6 Adjoint systems | 55 |

7 The residue theorem in its projective form | 61 |

9 The pluricanonical systems | 65 |

Appendix to Chapter III | 66 |

The Arithmetic Genus and the Generalized Theorem of RIEMANNROCH | 70 |

2 The theorem of RlEMANNROCH on algebraic surfaces | 72 |

3 The deficiency of the characteristic series of a complete linear system | 75 |

4 The elimination of exceptional curves and the characterization of ruled surfaces | 78 |

Appendix to Chapter IV | 83 |

Continuous Nonlinear Systems | 87 |

2 Complete continuous systems and algebraic equivalence | 90 |

3 The completeness of the characteristic series of a complete continuous system | 93 |

4 The variety of PICARD | 99 |

6 The theory of the base and the number ϱ of PICARO | 102 |

7 The division group and the invariant σ of SEVERI | 106 |

8 On the moduli of algebraic surfaces | 108 |

Appendix to Chapter V | 113 |

Topological Properties of Algebraic Surfaces | 124 |

3 Algebraic cycles on F and their intersections | 125 |

4 The representation of F upon a multiple plane | 126 |

5 The deformation of a variable plane section of F | 127 |

6 The vanishing cycles and the invariant cycles | 128 |

Simple and Double Integrals on an Algebraic Surface | 151 |

2 Simple integrals of the second kind | 152 |

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 | 154 |

4 The normal functions of POINCARÉ | 160 |

5 The existence theorem of LEFSCHETZPOINCARÉ | 164 |

6 Reducible integrals Theorem of POINCARÉ | 168 |

7 Miscellaneous applications of the existence theorem | 172 |

8 Double integrals of the first kind Theorem of HODGE | 177 |

9 Residues of double integrals and the reduction of the double integrals of the second kind | 181 |

10 Normal double integrals and the determination of the number of independent double integrals of the second kind | 186 |

Appendix to Chapter VII | 192 |

Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves | 202 |

2 Properties of the fundamental group G | 205 |

3 The irregularity of cyclic multiple planes | 206 |

4 Complete continuous systems of plane curves with d nodes | 209 |

5 Continuous systems of plane algebraic curves with nodes and cusps | 214 |

Appendix 1 to Chapter VIII | 219 |

Appendix 2 to Chapter VIII | 224 |

Series of Equivalence | 227 |

2 Series of equivalence | 228 |

3 Invariant series of equivalence | 230 |

4 Topological and transcendental properties of series of equivalence | 232 |

5 Added in 2nd edition by D Mumford | 233 |

Correspondences between Algebraic Varieties | 234 |

2 The transcendental equations and the rank of a correspondence | 236 |

3 The case of two coincident varieties Correspondences with valence | 239 |

4 The principle of correspondence of ZEUTHBNSEVERI | 240 |

Bibliography | 243 |

251 | |

Index | 264 |

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### Common terms and phrases

abelian adjoint algebraic algebraic curve algebraic surface algebraic varieties algebriche Amer arbitrary assigned assume base points birational branch called Chapter characteristic series characters coincide complete components condition consequently consider contained continuous system correspondence curves cycles defined definition deformation denote determined difference dimension distinct divisors double effective ENRIQUES equal equations equivalence exceptional curves exists fact finite fixed follows formula function fundamental genus given hence holds immediately independent infinitely integrals intersection invariant irreducible kind LEFSCHETZ linear system Math matrix multiple necessary non-singular normal obtained passing pencil periods PICARD plane positive possesses possible preceding projective proof proved rational reducible regular relation relative residue respectively result satisfy SEVERI shows simple singularities space superficie surface surface F theorem theory transformations variable varies variety virtual zero