[Paper Review] Resolution of Singularities for 3-folds in positive characteristic

Steven Dale Cutkosky|ArXiv.org|Jun 21, 2006

Algebraic Geometry and Number Theory28 references4 citations

TL;DR

This paper presents a concise, self-contained proof of resolution of singularities for 3-folds in positive characteristic greater than 5, using embedded resolution of surfaces and principalization of ideals via blow-ups along points and nonsingular curves. The key contribution is a streamlined algorithm based on order-based invariants and weak transforms, achieving resolution without relying on modern simplifications of Hironaka's method.

ABSTRACT

In this paper a concise, complete proof of resolution of singularities of 3-folds in positive characteristic (>5) is given. The first proof of this theorem was given by Abhyankar in 1966. The resolution morphism in our proof is an isomorphism over the nonsingular locus.

Motivation & Objective

• To provide a complete, concise proof of resolution of singularities for 3-folds in positive characteristic, specifically for fields of characteristic ≠ 2, 3, 5.
• To overcome the absence of hypersurfaces of maximal contact in positive characteristic, which hinders standard characteristic zero techniques.
• To establish a self-contained proof using only methods available by 1967, avoiding later developments.
• To simplify Abhyankar’s original 508-page proof by focusing on order-based invariants and weak transforms rather than multiplicity.
• To motivate further research on open problems in positive characteristic resolution, particularly embedded resolution of 3-folds in 4-folds.

Proposed method

• The proof relies on embedded resolution of surface singularities in a nonsingular 3-fold, ensuring the total transform is a simple normal crossings divisor.
• It uses principalization of ideal sheaves on nonsingular 3-folds via blow-ups of points and nonsingular curves where the ideal is not invertible.
• The algorithm follows Levi’s method for resolving 2D hypersurfaces, adapted to positive characteristic using Hironaka’s invariant with transversality to exceptional divisors.
• Key techniques include the use of weak transforms instead of strict transforms, which simplifies the invariant tracking by focusing on ideal order rather than multiplicity.
• Normalization and Zariski–Abhyankar factorization are used to resolve fundamental loci and ensure compatibility across blow-up sequences.
• The construction proceeds through successive blow-ups of points and curves, with careful control of the fundamental locus and transversality to exceptional divisors.

Experimental results

Research questions

• RQ1 Can resolution of singularities for 3-folds in positive characteristic be achieved with a self-contained, concise proof using only classical techniques?
• RQ2 How can the absence of hypersurfaces of maximal contact in positive characteristic be circumvented in resolution algorithms?
• RQ3 Can the resolution invariant be based on ideal order rather than multiplicity, leading to simpler proofs in positive characteristic?
• RQ4 Is it possible to achieve transversality with the exceptional divisor during blow-ups in positive characteristic without relying on characteristic zero tools?
• RQ5 What is the minimal set of assumptions on the base field (e.g., characteristic ≠ 2,3,5) required for resolution of 3-folds?

Key findings

• A complete resolution of singularities is achieved for 3-folds over algebraically closed fields of characteristic ≠ 2, 3, 5, resulting in a nonsingular projective variety W and a birational morphism φ: W → V.
• The resolution process is constructed via a sequence of blow-ups at points and nonsingular curves, ensuring the total transform of the singular locus and exceptional divisors forms a simple normal crossings divisor.
• The proof establishes embedded resolution of surfaces in a nonsingular 3-fold, with the strict transform of the surface and the total transform of the divisor being simple normal crossings.
• Principalization of ideal sheaves on nonsingular 3-folds is achieved through blow-ups along centers where the ideal is not invertible, resulting in locally principal ideals.
• The method ensures that the resolution morphism is an isomorphism above the nonsingular locus of the original variety V.
• The construction avoids reliance on modern simplifications of Hironaka’s proof, making it accessible with 1967-era techniques.