Algebraic Surfaces

The 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 Q]lotted 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 teKt differ, to a greater or less extent, from the proofs given in the original papers.The author regrets that he has not been able, for the reasons outlined above, to include in his work two interesting and important developments of the theory: (I) the classification of surfaces by means of their invariants, due chiefly to ENRIQUES; (II) the theory of real algebraic surfaces, due to CoMESSATTI. Fortunately, excellent and quite recent accounts of these two developments are available (1. GEPPERT, a; II. COMESSATTI, b; see "Bibliography"). Front Matter ....Pages I-V Theory and reduction of singularities (O. Zariski)....Pages 1-23 Linear systems of curves (O. Zariski)....Pages 23-44 Adjoint systems and the theory of invariants (O. Zariski)....Pages 44-63 The arithmetic genus and the generalized theorem of Riemann-Roch (O. Zariski)....Pages 63-76 Continuous non-linear systems (O. Zariski)....Pages 76-101 Topological properties of algebraic surfaces (O. Zariski)....Pages 101-119 Simple and double integrals on an algebraic surface (O. Zariski)....Pages 119-159 Branch curves of multiple planes and continuous systems of plane algebraic curves (O. Zariski)....Pages 160-175 Back Matter ....Pages 175-202