[Paper Review] Uniformisation locale des schémas quasi-excellents de caractéristique nulle

Jean-Christophe San Saturnino|arXiv (Cornell University)|Nov 14, 2013

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper establishes embedded local uniformization for valuations centered on quasi-excellent schemes of characteristic zero by reducing to rank-one valuations, desingularizing the ideal of elements with infinite valuation, and monomializing key polynomials. The key result is a monomialization theorem valid in all characteristics under the absence of limit key polynomials—a condition always satisfied in characteristic zero, thus proving local uniformization in this setting via key polynomial techniques and successive blow-ups.

ABSTRACT

We prove an embedded local uniformization theroem for a valuation centered on a point of a quasi-excellent scheme of characteristic zero. The proof reduces to valuations of rank 1 and consists in desingularizing the ideal formed by the elements of infinite value and monomializing the key polynomials. We then prove a monomialization theorem valid in all characteristic under certain conditions, including that of non-existence of limit key polynomials, a condition that is always satified in characteristic zero.

Motivation & Objective

• To establish an embedded local uniformization theorem for valuations centered on quasi-excellent schemes of characteristic zero.
• To reduce the problem to rank-one valuations via reduction techniques from the literature.
• To resolve the ideal of elements with infinite valuation in the completion of the local ring.
• To monomialize key polynomials through a sequence of local blow-ups preserving monomial forms.
• To prove a monomialization theorem valid in all characteristics under the condition of no limit key polynomials, which holds in characteristic zero

Proposed method

• Reduces the problem to rank-one valuations using a known reduction result (Novacoski–Spivakovsky, 2012).
• Identifies the ideal $ar{H}$ of elements with infinite valuation in the completion $ ilde{S}$, which is prime and called the implicit prime ideal.
• Uses the quasi-excellence of $S$ to deduce that $ ilde{S}_{ar{H}}$ is regular, so the problem reduces to regularity and monomialization of $ ilde{S}/ar{H}$.
• Applies Cohen’s structure theorem to represent $ ilde{S}/ar{H}$ as a quotient of a regular complete local ring $R$, and works on $R/H$.
• Employs induction on the embedding dimension, using key polynomials to guide blow-ups that preserve monomial forms.
• Demonstrates that in characteristic zero, no limit key polynomials exist (Corollary 3.13), enabling finite-step monomialization via recursive blow-ups and translations.

Experimental results

Research questions

• RQ1 Can embedded local uniformization be achieved for valuations centered on quasi-excellent schemes of characteristic zero using key polynomial techniques?
• RQ2 Under what conditions does monomialization of key polynomials succeed in arbitrary characteristic, particularly in the absence of limit key polynomials?
• RQ3 Does the non-existence of limit key polynomials in characteristic zero ensure finite-step monomialization and hence local uniformization?
• RQ4 Can the structure of the completion and the implicit prime ideal $ar{H}$ be used to reduce the uniformization problem to monomialization of a quotient ring?
• RQ5 How do successive local blow-ups preserve monomial forms when applied to key polynomials, and what guarantees termination in characteristic zero?

Key findings

• The ideal $ar{H}$ of elements of infinite valuation in the completion $ ilde{S}$ is prime and the localization $ ilde{S}_{ar{H}}$ is regular when $S$ is quasi-excellent.
• The absence of limit key polynomials in characteristic zero (Corollary 3.13) ensures that monomialization can be achieved in finitely many steps.
• A sequence of local blow-ups can be constructed such that all key polynomials become monomials, and once monomialized, they remain so under further blow-ups.
• The monomialization process is effective and constructive: it yields a sequence of blow-ups that regularize $ ilde{S}/ar{H}$ and monomialize all elements via key polynomial decomposition.
• The main result is a proof of embedded local uniformization for quasi-excellent schemes of characteristic zero, relying on the non-existence of limit key polynomials.
• The method provides a complete description of all infinitely near points and intermediate blow-ups in advance, offering a global control over the resolution process.