[Paper Review] Resolution of singularities of an idealistic filtration in dimension 3 after Benito-Villamayor

Hiraku Kawanoue, Kenji Matsuki|arXiv (Cornell University)|May 21, 2012

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper presents a local algorithm for resolving singularities of an idealistic filtration in dimension 3 over fields of positive characteristic, incorporating Benito-Villamayor's method into the Idealistic Filtration Program. It introduces a new invariant that strictly decreases with each blow-up, overcoming the Abhyankar-Moh pathology and 'Kangaroo' points, and establishes termination in the monomial case, completing the algorithm in dimension 3.

ABSTRACT

We establish an algorithm for resolution of singularities of an idealistic filtration in dimension 3 (at the local level) in positive characteristic, incorporating the method recently developed by Benito-Villamayor into our framework. Although (a global version of) our algorithm only implies embedded resolution of surfaces in the smooth ambient space of dimension 3, a classical result known before, we introduce some new invariant which effectively measures how much singularities are improved in the process of our algorithm and which strictly drops after each blow up. This is in contrast to the well-known Abhyankar-Moh pathology of the increase of the residual order under blow up and the phenomenon of the "Kangaroo" points observed by Hauser.

Motivation & Objective

• To develop a local algorithm for resolution of singularities of an idealistic filtration in dimension 3 over positive characteristic fields.
• To incorporate Benito-Villamayor's recent method into the Idealistic Filtration Program framework.
• To construct a new invariant that strictly decreases under blow-ups, countering the Abhyankar-Moh pathology and 'Kangaroo' points.
• To solve the monomial case in dimension 3, thereby completing the algorithm in this setting.
• To lay the foundation for embedded resolution of 3-folds in nonsingular 4-folds in positive characteristic.

Proposed method

• The algorithm proceeds by induction on a new invariant σ, replacing the classical hypersurface of maximal contact used in characteristic zero.
• A new strand of invariants, denoted inv_new, is woven into the process to measure singularity improvement and ensure strict decrease after each blow-up.
• The method distinguishes between cases based on the invariant τ, with special treatment for τ = 1 using a 'cleaning' procedure and the invariant H.
• The monomial case in dimension 3 is analyzed in detail through case-by-case analysis of τ, with explicit procedures and termination proofs.
• The framework uses idealistic filtration of i.f.g.-type and applies D-saturation and R-saturation to ensure compatibility across local patches.
• The global version is constructed via patching of local filtrations, relying on Hilbert-Samuel function maximality and persistence under transformations.

Experimental results

Research questions

• RQ1 Can a local algorithm for resolution of singularities of an idealistic filtration be constructed in dimension 3 over positive characteristic fields?
• RQ2 How can a new invariant be defined such that it strictly decreases under blow-ups, countering the Abhyankar-Moh pathology?
• RQ3 What is the structure and behavior of the monomial case in dimension 3 under the new invariant framework?
• RQ4 Can the method be extended to yield embedded resolution of surfaces in nonsingular 3-folds over positive characteristic fields?
• RQ5 Does the algorithm provide a viable path toward embedded resolution of 3-folds in nonsingular 4-folds in positive characteristic?

Key findings

• The paper constructs a new invariant, inv_new, which strictly decreases after each blow-up, ensuring termination and effective singularity improvement.
• The monomial case in dimension 3 is fully resolved through detailed case analysis based on τ, with a dedicated 'cleaning' procedure for τ = 1.
• The algorithm successfully overcomes the Abhyankar-Moh pathology and 'Kangaroo' points by using a new invariant that tracks improvement.
• The global version of the algorithm yields embedded resolution of surfaces in nonsingular 3-folds, recovering a classical result via a new method.
• The method provides a foundational step toward embedded resolution of 3-folds in nonsingular 4-folds in positive characteristic.
• The idealistic filtration is constructed via D-saturation and R-saturation, ensuring compatibility and persistence of the singular locus under transformations.