[Paper Review] Abhyankar places admit local uniformization in any characteristic

Hagen Knaf, Franz‐Viktor Kuhlmann|arXiv (Cornell University)|Apr 12, 2003

Algebraic Geometry and Number Theory6 references4 citations

TL;DR

This paper establishes local uniformization for Abhyankar places in arbitrary characteristic by leveraging valuation-theoretic methods. It proves that $K$-trivial Abhyankar places with separable residue field extensions admit local uniformization on smooth varieties without field extensions, and extends the result to non-trivial places over regular Nagata rings of dimension at most 2 under defectless, torsion-free, and separable conditions.

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 $FP|K$ is separable. We generalize this result to the case where $P$ dominates a regular local Nagata ring $R\subseteq K$ of Krull dimension $\dim R\leq 2$, 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 $FP|KP$ is separable. The results also include a form of monomialization. Further, we show that in both cases, finitely many Abhyankar places admit simultaneous local uniformization on an affine scheme if they have value groups isomorphic over $v_P K$.

Motivation & Objective

• To generalize Zariski's local uniformization theorem to arbitrary characteristic by focusing on Abhyankar places.
• To resolve the open problem of local uniformization for Abhyankar places when the residue field extension is separable.
• To extend the result to places dominating regular local Nagata rings of dimension ≤ 2, under defectless, torsion-free, and separable conditions.
• To provide a monomialization form, showing that elements can be expressed as monomials in regular parameters.
• To prove simultaneous local uniformization for finitely many Abhyankar places with isomorphic value groups over $v_P K$.

Proposed method

• Use valuation-theoretic techniques to analyze the structure of Abhyankar places in function fields over arbitrary fields.
• Embed the function field $F$ into the fraction field of the strict henselization of $\mathcal{O}_K(T)$ for a transcendence basis $T$.
• Apply the notion of $R$-uniformizability for pairs $(P, Z)$, where $Z$ is a finite set of elements in $\mathcal{O}_P$.
• Utilize the condition that $\dim \mathcal{O}_{X,x} = \dim_{\mathbb{Q}}(v_P F / v_P K \otimes \mathbb{Q}) + 1$ or $+2$ depending on the base ring $R$.
• Ensure that all elements in $Z$ become $\mathcal{O}_{X,x}$-monomials in a regular system of parameters of $\mathcal{O}_{X,x}$.
• Apply the theory of defectless valued fields and universal catenarity to ensure the existence of regular models over $R$.

Experimental results

Research questions

• RQ1 Can local uniformization be achieved for Abhyankar places in positive characteristic without field extensions?
• RQ2 Under what conditions on the value group and residue field extension does local uniformization hold for non-$K$-trivial places?
• RQ3 Can multiple Abhyankar places with isomorphic value groups be simultaneously uniformly resolved?
• RQ4 What is the precise dimension of the local ring at the center of a place under uniformization in the non-trivial base ring case?
• RQ5 How can monomialization be achieved in the context of local uniformization for Abhyankar places?

Key findings

• Every $K$-trivial Abhyankar place with separable residue field extension admits local uniformization on a smooth variety without extending the function field.
• For non-trivial Abhyankar places over a regular local Nagata ring $R$ of dimension $\leq 2$, local uniformization holds if $(K,P)$ is defectless, $v_P F / v_P K$ is torsion-free, and $FP/KP$ is separable.
• The dimension of the local ring $\mathcal{O}_{X,x}$ at the center $x$ of the place satisfies $\dim \mathcal{O}_{X,x} = \dim_{\mathbb{Q}}(v_P F / v_P K \otimes \mathbb{Q}) + 1$ if $\mathcal{O}_K$ is a discrete valuation ring, and $+2$ otherwise.
• All elements in a given finite set $Z \subset \mathcal{O}_P$ can be expressed as monomials in a regular system of parameters of $\mathcal{O}_{X,x}$.
• Finitely many Abhyankar places with isomorphic value groups over $v_P K$ admit simultaneous local uniformization on a single affine scheme.
• The base ring $R$ is necessarily Nagata, and the defectless condition ensures the integrality of the integral closure of $R/p$ in finite extensions.