Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-27T01:13:05.717Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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