Annali di Matematica Pura ed Applicata

, Volume 29, Issue 1, pp 187–198 | Cite as

A fundamental lemma from the theory of holomorphic functions on an algebraic variety

  • Oscar Zariski


Holomorphic Function Algebraic Variety Fundamental Lemma 
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  1. (1).
    A brief summary of this unpublished work has appeared in theAbstracts of addresses given at the Conference on Algebraic Geometry and Algebraic Number Theory, the University of Chicago, the Department of Mathematics, January, 1949. In the abstract of our address (Theory and applications of holomorphic functions on algebraic varieties), given at that conference, we also present the principal application of the theory of holomorphic functions, namely the derivation of the « principle of degeneration » in abstract algebraic geometry. Some of the ground work for this theory has been laid in the following papers of ours: (1)Generalized semi-local rings, « Summa Brasiliensis Mathematicae », Vol. I, fasc. 8, 1946. (2)Analytical irreducibility of normal varieties, « Ann. of Math. », vol. 49, 1948. (3)A simple analytical proof of a fundamental property of birational transformations, « Proc. Mat. Acad. Sci. », vol. 35, 1949.Google Scholar
  2. (2).
    See our paperThe compactness of the Riemann manifold of an abstract of algebraic functions, « Bull. Amer. Math. Soc. », vol. 40, 1044.Google Scholar
  3. (4).
    See our paperThe concept of a simple point of an abstract algebraic variety, « Trans-Amer. Math. Soc. », vol. 62, 1947, p. 12, section 3. 3. This paper will be referred to in the sequel as SP.Google Scholar

Copyright information

© Swets & Zeitlinger B. V. 1949

Authors and Affiliations

  • Oscar Zariski
    • 1
  1. 1.CambridgeUSA

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