[Paper Review] Resolution of Singularities of Germs in Characteristic Positive Associated with Valuation Rings of Iterated Divisor Type

Tohsuke Urabe|arXiv (Cornell University)|Jan 11, 1999

Algebraic Geometry and Number Theory6 references6 citations

TL;DR

This paper presents a resolution of singularities for hypersurface germs in positive characteristic using iterated monoidal transformations centered on smooth subvarieties, provided a valuation ring of iterated divisor type is associated with the germ. The key contribution is a framework involving space germs, Weierstrass representations, and reduction sequences that enables resolution under this condition, though the paper was later withdrawn by the author.

ABSTRACT

In this paper we show that any hypersurface singularities of germs of varieties in positive characteristic can be resolved by iterated monoidal transformations in centers in smooth subvarieties, if we have a valuation ring of iterated divisor type associated with the germ. Besides, we introduce fundamental concepts for the study of resolution of singularities of germs such as space germs, iterated analytic monoidal transformations with a normal crossing, Weierstrass representations, reduction sequences, and so forth.

Motivation & Objective

• To develop a resolution method for hypersurface singularities in positive characteristic when a valuation ring of iterated divisor type is associated with the germ.
• To introduce foundational concepts such as space germs, iterated analytic monoidal transformations with normal crossings, and Weierstrass representations.
• To define reduction sequences as a tool for tracking the resolution process in positive characteristic.
• To establish a general framework that extends to broader cases beyond the special setting studied.
• To provide structural tools and insights that could support future resolution in positive characteristic, despite the paper's withdrawal.

Proposed method

• Utilizes iterated monoidal transformations centered on smooth subvarieties to resolve singularities of hypersurface germs.
• Employs valuation rings of iterated divisor type as a key structural assumption to guide the resolution process.
• Introduces the concept of space germs to formalize local analytic behavior around singular points.
• Applies Weierstrass representations to express local equations in a form amenable to transformation and simplification.
• Uses reduction sequences to systematically track changes in the singularity during successive transformations.
• Implements normal crossing conditions in iterated monoidal transformations to ensure controlled behavior of exceptional divisors.

Experimental results

Research questions

• RQ1 Can hypersurface singularities in positive characteristic be resolved using iterated monoidal transformations when a valuation ring of iterated divisor type is associated with the germ?
• RQ2 What structural tools are necessary to generalize resolution techniques from characteristic zero to positive characteristic?
• RQ3 How can Weierstrass representations and reduction sequences be used to control the resolution process in positive characteristic?
• RQ4 What role do space germs and normal crossing conditions play in enabling resolution via monoidal transformations?
• RQ5 In what way do valuation rings of iterated divisor type serve as a bridge to resolving singularities in positive characteristic?

Key findings

• The paper establishes that resolution of hypersurface germs in positive characteristic is achievable via iterated monoidal transformations when a valuation ring of iterated divisor type is associated with the germ.
• The framework introduced—featuring space germs, Weierstrass representations, and reduction sequences—provides a systematic approach to tracking singularities during resolution.
• The method relies on iterated monoidal transformations with normal crossing conditions to maintain control over the exceptional divisors.
• The paper claims to contain not only a result in a special case but also foundational tools applicable to broader resolution problems.
• Despite its theoretical contribution, the paper was withdrawn by the author, indicating possible unresolved issues or revisions in the final version.