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A result relating to the local uniformisation theorem

Published online by Cambridge University Press:  26 February 2010

J. Herszberg
J. Herszberg
Affiliation:
Birkbeck College, London, W.C.I.
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The Local Uniformisation Theorem was proved by O. Zariski [5] in 1940, and, for the general case, it is so far the only existing proof of the theorem.

Let V be an irreducible manifold defined over a ground field of characteristic zero, and let Σ be its function field. Suppose V lies in an affine space An, and D is any subvariety of V not at infinity. Let J be the integral domain of V and ρ be the prime ideal in J defining D. Then we denote the quotient ring of D by Q(D|V), and by this we shall mean the quotient ring Jρ [1; p. 99]. Thus when we deal with subvarieties of two birationally equivalent manifolds V and V', then the quotient rings will always be subrings of the same representation of the function field of V and V'. Let B be any valuation of Σ whose centre on V is C. The Local Uniformisation Theorem states that there exists a birational transform V' of V such that the centre C' of B on V' is simple and Q(C/V) ⊆Q(C'/V').

Type
Research Article
Information
Mathematika , Volume 9 , Issue 2 , December 1962 , pp. 104 - 110
Copyright
Copyright © University College London 1962

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References

1. Hodge, W. V. D. and Pedoe, D., Methods of algebraic geometry, Vol. 3, Ch. 18 (Cambridge, 1954).Google Scholar
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5. Zariski, O., “Local uniformization on algebraic varieties”, Annals of Math., 41 (1940), 852–896.Google Scholar