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A note on uniform definability and minimal fields of definition

Published online by Cambridge University Press:  12 March 2014

Rahim Moosa*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801., USA, E-mail: [email protected]

Extract

Let k ⊂ K be a field extension, where K is an algebraically closed field of any characteristic and k is the prime field. Recall the following property of Hilbert Schemes (see, for example, [1], Proposition 1.16): Suppose × S is a flat family of closed subschemes of parametrised by a scheme S/k. Then for every closed subscheme Z in , if [Z] denotes the Hilbert point of Z in Hilb() then the residue field of Hilb() at [Z] is the minimal field of definition for Z. Intuitively, this says that as a family parametrised by Hilb(), each fibre of lies above a point whose “co-ordinates” generate its minimal field of definition.

In the following note we are concerned with a more naive form of the above situation, for which we wish to give an elementary account. Suppose ϕ(x,y) is a system of polynomial equations over k (in variables x = (x1,…, xm) and parameters y = (y1, …, yn)), such that

is a family of (possibly reducible) affine varieties in Km. Each nonempty member of this family has a unique minimal field of definition. The question arises as to whether it is possible to express this family of varieties using parameters that come, pointwise, from these minimal fields of definition. That is, is there a system of polynomial equations ψ(x, z) over k, such that each ψ(x, b) with bKN is of the form Va for some aKn; and such that each Va is defined by ψ(x, b) for some bKN whose coordinates generate the minimal field of definition for Va? Moreover, we would like b to be obtained definably from a.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Kollár, J., Rational curves on algebraic varieties, Springer-Verlag, 1991.Google Scholar
[2]Lang, S., Introduction to algebraic geometry, Interscience, 1958.Google Scholar
[3] van den Dries, L. and Schmidt, K., Bounds in the theory of polynomial rings over fields: A nonstandard approach, Inventiones Mathematicae, vol. 76 (1984), pp. 7791.CrossRefGoogle Scholar