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Bases for the Prime Ideals Associated with Certain Classes of Algebraic Varieties

Published online by Cambridge University Press:  24 October 2008

L. S. Goddard
L. S. Goddard
Affiliation:
St John's CollegeCambridge
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Extract

The fact that the prime ideal associated with a given irreducible algebraic variety has a finite basis is a pure existence theorem. Only in a few isolated particular cases has the base for the ideal been found, and there appears to be no general method for determining the base which can be carried out in practice. Hilbert, who initiated the theory, proved that the prime ideal defining the ordinary twisted cubic curve has a base consisting of three quadrics, and contributions to the ideal theory of algebraic varieties have been made by König, Lasker, Macaulay and, more recently, by Zariski. A good summary, from the viewpoint of a geometer, is given by Bertini [(1), Chapter XII]. However, the tendency has been towards the development of the pure theory. In the following paper we actually find the bases for the prime ideals associated with certain classes of algebraic varieties. The paper falls into two parts. In Part I there is proved a theorem (the Principal Theorem) of wide generality, and then examples are given of some classes of varieties satisfying the conditions of the theorem. In Part II we find the base for the prime ideals associated with Veronesean varieties and varieties of Segre. The latter are particularly interesting since they represent (1, 1), without exception, the points of a multiply-projective space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 39 , Issue 1 , March 1943 , pp. 35 - 48
Copyright
Copyright © Cambridge Philosophical Society 1943

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References

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