## 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

I | 4 |

II | 9 |

III | 11 |

IV | 12 |

V | 16 |

VI | 23 |

VIII | 26 |

IX | 28 |

LI | 134 |

LII | 136 |

LIII | 137 |

LIV | 138 |

LV | 139 |

LVI | 141 |

LVII | 142 |

LVIII | 146 |

X | 30 |

XI | 33 |

XII | 35 |

XIII | 40 |

XIV | 42 |

XV | 44 |

XVI | 50 |

XVIII | 52 |

XIX | 54 |

XX | 57 |

XXI | 59 |

XXII | 65 |

XXIV | 69 |

XXV | 70 |

XXVI | 74 |

XXIX | 76 |

XXX | 79 |

XXXI | 82 |

XXXII | 87 |

XXXIII | 91 |

XXXV | 94 |

XXXVI | 97 |

XXXVII | 103 |

XXXIX | 106 |

XL | 110 |

XLI | 112 |

XLII | 117 |

XLIII | 128 |

XLVII | 129 |

XLVIII | 130 |

XLIX | 131 |

L | 132 |

LIX | 155 |

LXII | 156 |

LXIII | 158 |

LXIV | 164 |

LXV | 168 |

LXVI | 172 |

LXVII | 176 |

LXVIII | 181 |

LXIX | 185 |

LXX | 190 |

LXXI | 196 |

LXXII | 206 |

LXXV | 209 |

LXXVI | 210 |

LXXVII | 213 |

LXXVIII | 218 |

LXXIX | 223 |

LXXX | 228 |

LXXXI | 231 |

LXXXIII | 232 |

LXXXIV | 234 |

LXXXV | 236 |

LXXXVI | 237 |

LXXXVII | 238 |

XC | 240 |

XCI | 243 |

XCII | 244 |

XCIII | 247 |

XCIV | 255 |

XCV | 268 |

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### Common terms and phrases

1-cycles abelian integral abelian varieties adjoint curves adjoint surfaces adjoint system algebraic curve algebraic cycles algebraic geometry algebraic surface algebraic varieties Amer arbitrary Atti Accad base points birational transformations birationally branch curve canonical CASTELNUOVO characteristic series cohomology complete continuous system condition correspondence curves of order cusps defined deformation denote dimension divisors double integrals double points ENRIQUES equations exceptional curves exists finite number fixed component follows formula fundamental curves genus given hence homologies hyperplane section independent infinitely intersection irreducible curve LEFSCHETZ linear system linearly equivalent Math matrix moduli multiple points nodes non-singular pencil periods PICARD plane section polynomials possesses proof proved reducible Rend residue ruled surfaces second kind section of F series of equivalence SEVERI simple integrals singularities subadjoint sufficiently high superficie algebriche surface F system of curves tangent theorem theory topological valence variable virtual multiplicities ZARISKI zero