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A result relating to the local uniformisation theorem

Published online by Cambridge University Press:  26 February 2010

J. Herszberg
Affiliation:
Birkbeck College, London, W.C.I.
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Extract

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
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
2. Northcott, D. G., “On the local cone of a point on an algebraic variety”, Journal London Math. Soc., 29 (1954), 326333.Google Scholar
3. Segre, B., “Sullo scioglimento delle singolarita delle varieta algebriche”, Ann. di Mat. (4), 33 (1952), 548.Google Scholar
4. Segre, B., “Dilatazioni e comportamenti associati nel campo analitico”, Rend. Mat. di Palermo (2), 1 (1952), 17.Google Scholar
5. Zariski, O., “Local uniformization on algebraic varieties”, Annals of Math., 41 (1940), 852896.Google Scholar