[Paper Review] Resolution of Singularities

Edward Bierstone, Pierre D. Milman|ArXiv.org|Sep 25, 1997

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper presents a constructive, canonical algorithm for resolving singularities in algebraic varieties over characteristic zero fields, using a novel local invariant to guide successive blow-ups along maximum locus centers. The method ensures desingularization through explicit, computable equations and provides a foundational framework for resolution in algebraic geometry.

ABSTRACT

This article is an exposition of an elementary constructive proof of canonical resolution of singularities in characteristic zero, presented in detail in Invent. Math. 128 (1997), 207-302. We define a new local invariant and get an algorithm for canonical desingularization by successively blowing up its maximum loci. The invariant can be described by local computations that provide equations for the centres of blowing up. We describe the origin of our approach and present the proof (in the hypersurface case) in parallel with a worked example.

Motivation & Objective

• To provide a constructive and canonical algorithm for resolving singularities in algebraic varieties over fields of characteristic zero.
• To define a new local invariant that captures the severity of singularities and guides the desingularization process.
• To demonstrate the algorithm in the hypersurface case with a detailed, worked example for clarity and verification.
• To establish a framework that is both elementary and rigorous, suitable for generalization and implementation.
• To offer a systematic method for computing centers of blow-ups using local equations derived from the invariant.

Proposed method

• Introduce a new local invariant that measures the complexity of singularities at each point of a variety.
• Use the invariant to define the centers of blow-ups as the loci where it achieves its maximum value.
• Apply a sequence of blow-ups along these centers, ensuring the invariant strictly decreases at each step.
• Construct the invariant via local computations that yield explicit equations for the centers of blow-ups.
• Prove that the process terminates in finitely many steps with a nonsingular variety.
• Present the method in parallel with a detailed example in the hypersurface case to illustrate the algorithm’s mechanics.

Experimental results

Research questions

• RQ1 How can a canonical desingularization algorithm be constructed in characteristic zero using a well-defined, computable invariant?
• RQ2 What is the structure of the local invariant that ensures termination and maximality in blow-up center selection?
• RQ3 Can the algorithm be made constructive and explicit through local equations for the centers of blow-ups?
• RQ4 How does the invariant behave under blow-ups, and what guarantees its strict decrease?
• RQ5 What is the role of the hypersurface case in illustrating the general method and validating the approach?

Key findings

• The proposed algorithm achieves canonical resolution of singularities in characteristic zero via a finite sequence of blow-ups.
• The new local invariant provides a precise, computable criterion for selecting centers of blow-ups, ensuring maximality and termination.
• The method is constructive and yields explicit equations for the centers, making it suitable for algorithmic implementation.
• The algorithm is proven to terminate in finitely many steps, with the invariant strictly decreasing at each blow-up.
• The approach is validated through a detailed example in the hypersurface case, demonstrating correctness and clarity.
• The framework generalizes to arbitrary varieties and provides a foundation for further developments in resolution theory.