Logarithmic resolution via multi-weighted blow-ups

This paper introduces the concept of multi-weighted blow-ups to construct an explicit, functorial algorithm for resolving singularities of reduced closed subschemes in characteristic zero by iteratively blowing up the worst singular locus until the result becomes a simple normal crossing divisor.

The Gist

Here is an explanation of the paper "Logarithmic resolution via multi-weighted blow-ups" by Dan Abramovich and Ming Hao Quek, translated into everyday language with creative analogies.

The Big Picture: Smoothing Out a Crumpled Map

Imagine you have a piece of paper (a mathematical space) that is crumpled, torn, and has sharp, jagged edges. In mathematics, this is called a singularity. The goal of this paper is to find a perfect, step-by-step recipe to smooth out that paper until it is perfectly flat and round, without tearing it further or losing any of its original shape.

This process is called Resolution of Singularities. It's like trying to iron out a wrinkled shirt, but the shirt is made of complex, multi-dimensional fabric, and you can't just use a hot iron; you have to use a very specific, mathematical set of tools.

The Problem: The "Worst" Spot First

For decades, mathematicians have known how to smooth these shapes, but the old methods were like trying to fix a broken vase by randomly hitting it with a hammer. They worked, but they were messy, slow, and sometimes depended on how you looked at the vase.

The authors of this paper want a perfect, automatic recipe.

• The Rule: Always fix the "worst" part of the crumple first.
• The Challenge: How do you define "worst"? And how do you fix it without making a mess elsewhere?

The New Tool: The "Multi-Weighted Blow-Up"

The authors introduce a new tool called a Multi-Weighted Blow-Up. To understand this, let's use an analogy.

Imagine you are sculpting a block of clay that has a deep, jagged crack in it.

1. Standard Blow-Up: You take a knife and slice right through the crack, then pull the two sides apart. This creates a new surface (an "exceptional divisor") where the crack used to be. It's a bit like opening a book; the pages are now visible.
2. Weighted Blow-Up: Sometimes, the crack is deeper on one side than the other. A standard slice isn't fair. A "weighted" blow-up is like using a specialized tool that stretches the clay more on the deep side and less on the shallow side, so the new surface is balanced.
3. Multi-Weighted Blow-Up (The Innovation): This is the paper's secret sauce. Imagine the crack isn't just deep or shallow; it's twisted, spiraling, and has different weights in different directions. A standard tool fails here. The Multi-Weighted Blow-Up is like a 3D printer that can apply different amounts of pressure in different directions simultaneously. It doesn't just stretch the clay; it reshapes the entire neighborhood of the crack according to a complex, multi-directional map.

The "Logarithmic" Twist: The Invisible Grid

The paper also uses something called Logarithmic Geometry.

• The Analogy: Imagine your clay sculpture is sitting on a grid of invisible strings (a "logarithmic structure"). These strings represent the "edges" or "boundaries" of the shape.
• Why it matters: In the old methods, when you smoothed the clay, you often accidentally cut these invisible strings or messed up the grid.
• The Solution: The authors' method respects the grid. They treat the invisible strings as part of the sculpture. When they smooth the clay, they ensure the strings remain intact and neatly arranged (what mathematicians call "Simple Normal Crossing"). This keeps the "map" of the shape consistent throughout the process.

The Algorithm: A Step-by-Step Recipe

Here is how their algorithm works, step-by-step:

1. Measure the Damage: They assign a "score" to every point on the crumpled paper. This score is a list of numbers (an invariant) that tells them exactly how bad the crumple is.
   • Analogy: Think of it like a weather report for the paper. "Here, the wind is 100 mph; there, it's 50 mph."
2. Find the Worst Spot: They look for the point with the highest "wind speed" (the worst singularity).
3. Apply the Multi-Weighted Tool: They use their special Multi-Weighted Blow-Up tool on that specific spot. Because the tool is so precise, it immediately lowers the "wind speed" (improves the singularity).
4. Repeat: They check the paper again. The worst spot is now gone, but a new (less bad) spot might have appeared. They repeat the process.
5. The Finish Line: Eventually, the "wind speed" drops to zero. The paper is perfectly smooth. The "invisible strings" (the logarithmic structure) are now perfectly aligned, forming a neat grid.

Why "Stacks" and "Artin"?

You might wonder why the paper mentions "Artin stacks" and "Deligne-Mumford stacks."

• The Analogy: Imagine you are trying to fix a crowd of people. If you just look at the crowd as a single blob, you miss the individuals. If you look at them as separate people, you miss the group dynamics.
• Stacks: A "stack" is a mathematical way of looking at a shape that remembers how it was built. It keeps track of the "symmetries" (like how a snowflake looks the same if you rotate it).
• The Benefit: By working with these "stacks," the authors can be more precise. They can smooth the shape without losing the memory of its symmetries. At the very end, if you want a simple shape (a "scheme"), you can peel away the extra "stack" layers, but the smoothing process was much easier because you kept those layers during the work.

The Result: A Universal Fix

The paper proves that this method:

1. Always Works: It will eventually smooth out any crumpled shape in characteristic zero (which includes all the numbers we use in everyday life).
2. Is Automatic: You don't need to make choices. The math decides the next step.
3. Is Consistent: If you have two identical shapes, the method will treat them exactly the same way, even if you look at them from different angles.

Summary

In short, Abramovich and Quek have invented a super-precise, multi-directional smoothing tool that respects the hidden grid of a shape. They use this tool to automatically and systematically iron out the worst wrinkles in complex mathematical shapes, ensuring that the final result is perfectly smooth and that the "map" of the shape remains clear and organized. It's a major upgrade from the old, clumsy hammers of the past.

Technical Summary

Here is a detailed technical summary of the paper "Logarithmic resolution via multi-weighted blow-ups" by Dan Abramovich and Ming Hao Quek.

1. Problem Statement

The paper addresses the problem of functorial resolution of singularities for a reduced, closed subscheme $X$ inside a smooth scheme $Y$ over a field of characteristic zero. The authors aim to achieve a resolution that satisfies two specific, often conflicting, requirements:

1. Hironaka-style Resolution: The process should be functorial with respect to smooth morphisms and resolve singularities by repeatedly blowing up the "worst" singular locus, eventually transforming the singular locus into a simple normal crossing (snc) divisor.
2. Logarithmic Geometry: The resolution should be compatible with logarithmic structures, specifically handling cases where the ambient space or the subscheme has toroidal (logarithmically smooth) singularities.

Previous approaches faced a trade-off:

• Classical algorithms (Hironaka, Bierstone-Milman, Encinas-Villamayor) work on schemes but often fail to produce an snc divisor for the singular locus without additional steps.
• Recent stack-theoretic approaches (Abramovich-Temkin-Włodarczyk, McQuillan) use weighted blow-ups on Deligne-Mumford stacks to achieve functoriality and immediate improvement of singularities. However, these often result in toroidal (logarithmically smooth) ambient spaces rather than smooth ones, requiring a separate "toroidal resolution" step at the end.
• Quek's previous work ([Que20]) used weighted toroidal blow-ups, which kept the process within logarithmic geometry but resulted in toroidal singularities rather than smooth ones.

The Goal: To construct an algorithm that stays within the realm of smooth ambient spaces (avoiding the need for a final toroidal resolution step) while maintaining functoriality and achieving logarithmic resolution, potentially at the cost of working with Artin stacks (which have non-trivial stabilizers) rather than schemes or Deligne-Mumford stacks.

2. Methodology

The authors introduce and utilize Multi-weighted blow-ups as the core mechanism for their algorithm.

A. Multi-Weighted Blow-ups

The authors refine the concept of weighted toroidal blow-ups using a construction by Satriano ([Sat13]).

• Definition: Given a monomial ideal $a$ on an affine space (or a toroidal scheme) and a vector of weights $b$, they construct a fantastack (a specific type of smooth toric Artin stack).
• Construction: This is realized as a quotient stack $[X_{\tilde{\Sigma}} / G_{\beta}]$, where $X_{\tilde{\Sigma}}$ is a toric variety associated with a refined fan, and $G_{\beta}$ is a diagonalizable group acting on it.
• Key Property: Unlike standard weighted blow-ups (which yield Deligne-Mumford stacks) or toroidal blow-ups (which yield toroidal stacks), multi-weighted blow-ups yield smooth Artin stacks. This allows the ambient space to remain smooth (in the stack-theoretic sense) throughout the process.

B. The Invariant and the "Worst" Locus

The algorithm relies on a sophisticated invariant, $\text{inv}_p(X \subset Y)$, defined at each point $p$.

• Structure: The invariant is a non-decreasing finite sequence of non-negative rational numbers (potentially ending in $\infty$). It is constructed inductively using:
  • Logarithmic order of the ideal.
  • Maximal contact elements.
  • Coefficient ideals.
  • Monomial saturation.
• Ordering: The set of invariants is well-ordered using a lexicographic order with a specific caveat: truncations of sequences are considered strictly larger than the full sequence. This ensures that the algorithm terminates.
• Center of Blow-up: The "worst singular locus" is defined as the closed substack where the invariant achieves its maximum value, $\max \text{inv}(X \subset Y)$.

C. The Algorithm

The resolution proceeds via a sequence of multi-weighted blow-ups:

1. Identify the center $C$ where $\text{inv}_p = \max \text{inv}$.
2. Perform the multi-weighted blow-up $\pi: Y' \to Y$ along the Rees algebra associated with this center.
3. Take the proper transform $X' \subset Y'$.
4. Result: The invariant strictly drops: $\max \text{inv}(X' \subset Y') < \max \text{inv}(X \subset Y)$.
5. Repeat until the invariant becomes $(1, 1, \dots, 1)$, indicating the subscheme is smooth and toroidal (or empty).

3. Key Contributions

1. Introduction of Multi-Weighted Blow-ups

The paper formalizes multi-weighted blow-ups as a distinct tool that bridges the gap between weighted toroidal blow-ups and standard weighted blow-ups.

• They are canonical in the sense of Satriano: they are the "canonical smooth, toroidal Artin stack" over the corresponding weighted toroidal blow-up.
• They allow the ambient space to remain smooth (as an Artin stack) rather than just toroidal.

2. Functorial Logarithmic Resolution

The authors prove Theorem A, establishing a canonical sequence of multi-weighted blow-ups that resolves the singularities of a reduced, generically toroidal subscheme $X \subset Y$.

• Properties:
  • The final ambient space $Y_N$ is a smooth Artin stack.
  • The final subscheme $X_N$ is smooth and toroidal.
  • The preimage of the singular locus is a simple normal crossing (snc) divisor.
  • The process is functorial with respect to strict, smooth morphisms of pairs.

3. Re-embedding Principle

Theorem C establishes that the resolution is independent of the embedding. If $X$ is embedded into a larger space $Y_1 = Y \times \mathbb{A}^1$, the resolution of $X$ in $Y_1$ is canonically identified with the resolution of $X$ in $Y$. This is crucial for patching local resolutions to form a global one.

4. Reduction to Schemes

While the output of the main algorithm is a smooth Artin stack, the authors show in Section 5.2 how to convert this into a smooth scheme. By applying Edidin-Rydh's reduction of stabilizers and Bergh-Rydh's destackification, one recovers the classical Hironaka resolution (Theorem 5.10). This demonstrates that the stack-theoretic approach is a powerful intermediate step that simplifies the construction of the classical scheme-theoretic resolution.

4. Main Results

• Theorem A (Logarithmic Embedded Resolution): For a reduced, generically toroidal closed substack $X$ of a smooth toroidal Artin stack $Y$, there exists a canonical sequence of multi-weighted blow-ups resolving $X$ such that the singular locus becomes an snc divisor.
• Theorem B (Single Step Improvement): A single multi-weighted blow-up along the "worst singular locus" strictly decreases the maximum invariant, ensuring progress toward resolution.
• Theorem D (Logarithmic Resolution): For any reduced, pure-dimensional Artin stack $X$, there exists a birational, surjective morphism to a smooth Artin stack $X'$ such that the exceptional locus is an snc divisor.
• Theorem 5.2 (Newton Non-degenerate Polynomials): For Newton non-degenerate polynomials, the algorithm terminates in a single step, generalizing known results for toric resolutions.

5. Significance

• Unification of Techniques: The paper successfully amalgamates the "worst locus" strategy of Hironaka/Bierstone-Milman with the stack-theoretic tools of Abramovich-Temkin-Włodarczyk and the logarithmic framework of Quek.
• Efficiency and Explicitness: The use of multi-weighted blow-ups provides a more explicit and efficient algorithm compared to previous methods. It avoids the need for a separate "toroidal resolution" phase at the end of the process.
• Handling of Stacks: By embracing Artin stacks as ambient spaces, the authors bypass the obstructions that prevent purely scheme-theoretic or Deligne-Mumford stack approaches from achieving both functoriality and smoothness simultaneously in the presence of logarithmic structures.
• Applications: The authors note that this framework is particularly useful for computations in algebraic geometry requiring logarithmic resolution, such as motivic integration and change of variables formulas for Artin stacks (referencing Satriano-Usatine).

In summary, this paper provides a robust, functorial, and efficient algorithm for resolving singularities in characteristic zero by leveraging the geometry of multi-weighted blow-ups on Artin stacks, effectively solving the problem of maintaining smoothness and logarithmic compatibility simultaneously.