Abhyankar places admit local uniformization in any characteristic

By H. Knaf, Franz-Viktor Kuhlmann. 2003

91 citations recorded by the source corpus.

Abstract We prove that every place P of an algebraic function field F | K of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field FP over K is equal to the transcendence degree of F | K , and the extension F P | K is separable. We generalize this result to the case where P dominates a regular local Nagata ring R ⊆ K of Krull dimension dim R ⩽ 2 , assuming that the valued field ( K , v P ) is defectless, the factor group v P F / v P K is torsion-free and the extension of residue fields F P | K P is separable. The results also include a form of monomialization.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9634e3af14728ec85d6b2cf51b3def3bb8d7a566

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

On fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop (2012, methodology): “This result is closely related to the work of Knaf-Kuhlmann (see [47]) proving a field-theoretic version of local uniformization for Abhyankar valuations of algebraic function fields, as well as to the work of M.”
Resolution of singularities of threefolds in positive characteristic. I.: Reduction to local uniformization on Artin–Schreier and purely inseparable coverings (2008, unknown): “Our formulation of theorem 7.2 stresses the role played by immediate wildly ramiﬁed extensions of valued ﬁelds which are the main obstruction to proving local uniformization, see [31] and [33].”
The algebra and model theory of tame valued fields (2013, unknown): “A C ] 1 4 Ju l 2 01 4”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, background): “rk(v) 6 n, and we say that v is an Abhyankar valuation if equality holds; such valuations can be geometrically characterized, see [36, 51, 48].”
Asymptotic Multiplicities of Graded Families of Ideals and Linear Series (2014, background): “Then ν is an “Abhyankar valuation”; that is trdegkk(X) = d = 0 + d = trdegkVν/mν + rational rank Γν , with k = Vν/mν , so there exists a local uniformization of ν by [26].”
RESOLUTION OF SINGULARITIES OF THREEFOLDS IN POSITIVE CHARACTERISTIC II (2008, background): “Our formulation of theorem 7.2 stresses the role played by immediate wildly ramified extensions of valued fields which are the main obstruction to proving local uniformization, see [ 31 ] and [33].”
Vanishing sequences and Okounkov bodies (2013, background): “By the main result of [16], Abhyankar valuations can be more explicitly characterized as quasimonomial valuations, i.”