×

Uniform approximation of Abhyankar valuation ideals in smooth function fields. (English) Zbl 1033.14030

Summary: Fix a rank one valuation \(\nu\) centered at a smooth point \(x\) on an algebraic variety over a field of characteristic zero. Assume that \(\nu\) 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 \(k>0\) such that \(a_{mn}\) is contained in \((a_{m-k})^n\) 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. The proof uses the theory of asymptotic multiplier ideals.

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13A15 Ideals and multiplicative ideal theory in commutative rings
14H05 Algebraic functions and function fields in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv