Uniform Approximation of Abhyankar Valuation Ideals in Smooth Function Fields

Details Uniform Approximation of Abhyankar Valuation Ideals in Smooth Function Fields Uniform Approximation of Abhyankar Valuation Ideals in Smooth Function Fields

2002-02-28

arxiv.org/abs/math/0202303v1

Mathematics

Algebraic Geometry

27 pages, latex

Scientific paper

In this paper we use the theory of multiplier ideals to show that the valuation ideals of a rank one Abhyankar valuation centered at a smooth point of a complex algebraic variety are approximated, in a quite strong sense, by sequences of powers of fixed ideals. Fix a rank one valuation v centered at a smooth point x on an algebraic variety over a field of characteristic zero. Assume that v is Abhyankar, that is, that its rational rank plus its transcendence degree equal the dimension of the variety. Let a_m denote the ideal of elements in the local ring of x whose valuations are at least m. Our main theorem is that there exists e>0 such that a_{mn} is contained in (a_{m-e})^n for all m and n. This can be viewed as a greatly strengthened form of Izumi's Theorem for Abhyankar valuations centered on smooth complex varieties.

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