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Valuation rings formed by ascending sequences of quotient rings

Published online by Cambridge University Press:  26 February 2010

J. Herszberg
J. Herszberg
Affiliation:
Birkbeck College, London, W.C.1.
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Extract

1. Certain geometric properties of the valuation theory were considered by O. Zariski in [7]. We have proved some related results in [1] and we consider further similar problems in this paper.

Let V be an irreducible algebraic primal situated in Sd, where d≥3. Throughout the ground field is the field K of complex numbers. For simplicity we assume that V lies in an affine space Ad with coordinates x1,…,xd. Let O be a point on V not at infinity and we take it to be the origin of Ad. Apply a monoidal transformation to V with O as the basis; We obtain thereby a (d−l)-fold V1 lying on a non-singular d-fold U1 situated in an affine space of dimension N1 Since V and V1 are birationally equivalent, we may identify their function fields and thus we denote their common function field by Σ.

Type
Research Article
Information
Mathematika , Volume 13 , Issue 1 , June 1966 , pp. 7 - 22
Copyright
Copyright © University College London 1966

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References

1. Herszberg, J., “Types of zero-dimensional valuations of function fields of algebraic manifolds with pre-assigned valuation rings”, Journal London Math. Soc., 35 (1960), 177–183.Google Scholar
2. Herszberg, J., “On a result of Beniamino Segre”, Rend, di Mat. Roma, V, vol. 19 (1960), 168–173.Google Scholar
3. Herszberg, J., “A result relating to the local uniformisation theorem”, Mathematika, 9 (1962), 104–110.Google Scholar
4. Hodge, W. V. D. and Pedoe, D., Methods of algebraic geometry, vol. 3 (Cambridge, 1954).Google Scholar
5. Lang, S., Introduction to algebraic geometry (Interscience, 1958).Google Scholar
6. Segre, B., “Dilatazioni e comportamenti associati nel campo analitico”, Rend. Mat. di Palermo (2), 1 (1952), 1–7.Google Scholar
7. Zariski, O., “Applicazioni geometriche della tooria dell valutazioni”, Rend di Mat. Roma, 13 (1954), 51–88.Google Scholar