Uniform approximation of Abhyankar valuation ideals in smooth function fields

By Lawrence Ein, R. Lazarsfeld, Karen E. Smith. 2002

156 citations recorded by the source corpus.

Fix a rank one valuation ν centered at a smooth point x on an algebraic variety over a field of characteristic zero. Assume that ν 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6646fbacabaff90c3de41d339936e632859cea69

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In Applied Mathematics

1. Mattias Jonsson
2. Chi Li
3. S. Boucksom
4. Yuchen Liu
5. H. Blum
6. M. Mustaţă
7. C. Favre
8. Ziquan Zhuang

In Computational Theory and Mathematics

1. Chi Li
2. Mattias Jonsson
3. M. Mustaţă
4. S. Boucksom
5. C. Favre
6. Yuchen Liu
7. R. Lazarsfeld
8. Craig Huneke

In Geometry and Topology

1. Chi Li
2. Mattias Jonsson
3. S. Boucksom
4. Yuchen Liu
5. H. Blum
6. M. Mustaţă
7. Robert J. Berman
8. C. Favre

Citations in the corpus

• Kähler–Einstein metrics and volume minimization (2016, background): “By [24], [49] and [18], the limitsup on the right hand side is a limit and is equal to the multiplicity of the graded family of ideals a• = {ap}:”
• Convex Bodies Associated to Linear Series (2008, background): “(10) The invariant mult(a•) was called the volume vol(a•) of a• in [18] and [31], but we prefer to stick with the terminology used in [29].”
• Asymptotic Multiplicities of Graded Families of Ideals and Linear Series (2014, background): “Theorem 6.5 is proven for valuation ideals associated to an Abhyankar valuation in a regular local ring which is essentially of finite type over a field by Ein, Lazarsfeld and Smith in [15], for general families of mR-primary ideals when R is a regular local ring containing a field by Mustaţă in [”
• Integral closure of ideals, rings, and modules (2006, methodology): “This has been done for example by Spivakovsky [277]; Cutkosky [52]; and Ein, Lazarsfeld and Smith [66].”
• The Valuative Tree (2002, methodology): “1the skewness is the inverse of the volume of a valuation as defined in [ ELS ] (see remark 8.5) 4 CHARLES FAVRE AND MATTIAS JONSSON”