Logarithmic and Weighted Resolution Algorithms

• Logarithmic and weighted resolution algorithms are systematic methods that use weighted blow-ups and log structures to canonically resolve singularities in algebraic geometry.
• They employ lexicographically ordered invariants and logarithmic orders to determine canonical centers, ensuring full functoriality under smooth and log-smooth maps.
• Empirical comparisons demonstrate that these techniques drastically reduce computational complexity, achieving factorial bounds and fewer blow-ups than classical methods.

A logarithmic and weighted resolution algorithm refers to a class of constructive, functorial procedures for principalization and resolution of singularities in algebraic geometry, exploiting the structure of logarithmic geometry and the algebraic machinery of weighted (often stack-theoretic) blow-ups. These methods achieve dramatic reductions in complexity compared to classical approaches via smooth blow-ups, offer fully canonical (history-free) algorithms, and are compatible with ambient logarithmic and stack-theoretic structures. The central principles were systematically developed by Abramovich, Temkin, Włodarczyk, McQuillan, Quek, Schober, and others, culminating in factorial-complexity algorithms and highly structured functorial correspondences in both embedded and non-embedded contexts (Brais, 1 Dec 2025, Abramovich et al., 2019, Abramovich et al., 17 Mar 2025, Temkin, 2023, Lee, 2020, Quek, 2020).

1. Weighted Blow-Ups and Logarithmic Structures

Weighted blow-ups generalize the classical blow-up construction by introducing a vector of positive integer (or rational) weights $w = (w_1,\dots,w_n)$ associated to regular system of parameters $(x_1,\dots,x_n)$ on a smooth affine variety $X = \Spec A$. The weight-filtration of $A$ is defined by

$$I_m := \langle x^a = x_1^{a_1}\cdots x_n^{a_n} : \sum_i w_i a_i \ge m \rangle \subset A,$$

and the associated Rees algebra

$I_w := \bigoplus_{m\ge0} I_m\, t^m \subset A[t].$

The weighted blow-up is then

$\widetilde X = \Proj I_w \longrightarrow X.$

For centers defined by monomials with fractional orders, stack-theoretic methods are necessary—affine charts become orbifold charts with group actions encoding the weights (Brais, 1 Dec 2025, Abramovich et al., 2019, Abramovich et al., 17 Mar 2025, Lee, 2020). Logarithmic geometry provides a framework where these weighted blow-ups extend to log schemes or fs log stacks with extra structure, enabling functorial treatment of boundary divisors and monomial ideals via log smoothness and Kummer-étale topology (Abramovich et al., 17 Mar 2025, Temkin, 2023, Quek, 2020).

2. Invariants, Centers, and Principalization Criteria

Central to the method is a functorial, local invariant that governs both the choice of center and the control of singularities:

• In the weighted regime, the invariant is a lexicographically ordered finite tuple of positive rationals $(a_1, a_2, \dots, a_k)$, defined recursively from the orders of vanishing and corresponding coefficient ideals (via repeated maximal contact reductions and factorial renormalizations) (Abramovich et al., 2019, Lee, 2020, Temkin, 2023).
• In the logarithmic context, the primary invariant is the “logarithmic order,” defined as

$\logord_x(I) = \min \{ d \mid D_X^{\le d}(I)_x = \O_{X, x} \},$

where $D_X^{\le d}$ is the sheaf of log differential operators of order $(x_1,\dots,x_n)$0 (Temkin, 2023, Abramovich et al., 17 Mar 2025).

• For hypersurfaces, polyhedral invariants such as the Newton polygon and the Bierstone–Milman invariant are used: the Newton graph’s faces dictate the primitive inward normals defining weighted centers; the initial forms along these faces control the reduction process (Brais, 1 Dec 2025).
• The canonical center is always the subscheme where the invariant achieves its maximum value, typically a locus defined by a weighted monomial ideal or a Kummer ideal in the log-geometry setting (Brais, 1 Dec 2025, Abramovich et al., 17 Mar 2025, Quek, 2020).

3. Algorithmic Procedure and Functoriality

Resolutions proceed algorithmically as follows:

1. Initialization: Given $(x_1,\dots,x_n)$1 or $(x_1,\dots,x_n)$2, where $(x_1,\dots,x_n)$3 is a simple normal crossings divisor and $(x_1,\dots,x_n)$4 the ideal to resolve.
2. Invariant Computation: At each step, compute the relevant invariant (lex tuple, log order, or Newton/Bierstone–Milman data).
3. Center Selection: Identify the canonical center—closure of the maximal-invariant locus—equipped with explicit weight data or log-structure (Brais, 1 Dec 2025, Abramovich et al., 2019, Abramovich et al., 17 Mar 2025).
4. Weighted Blow-Up: Perform the stack-theoretic weighted blow-up along the center. In codimension $(x_1,\dots,x_n)$5, quotient singularities arise, and so all constructions are done in the category of (log-)Deligne–Mumford stacks (Brais, 1 Dec 2025, Quek, 2020, Temkin, 2023).
5. Update Structures: Pull back the divisor, transform the ideal appropriately (using weak or strict transform), and enlarge the log-structure where required to maintain log smoothness (Brais, 1 Dec 2025, Abramovich et al., 17 Mar 2025).
6. Termination Check: Iterate until the invariant is minimal (i.e., only normal crossings remain), guaranteeing that the transform is principal/monomial and nonsingular (Abramovich et al., 2019, Abramovich et al., 17 Mar 2025, Lee, 2020).
7. Functoriality: All steps are canonical, no choices are made, and functoriality under (log-)smooth morphisms is strictly preserved; centers, invariants, and blow-ups pull back compatibly (Brais, 1 Dec 2025, Abramovich et al., 2019, Abramovich et al., 17 Mar 2025, Temkin, 2023, Quek, 2020).

4. Complexity and Comparative Efficiency

Weighted and logarithmic resolution algorithms offer substantial complexity improvements over classical methods. In the weighted case: $(x_1,\dots,x_n)$6 where $(x_1,\dots,x_n)$7 is the ambient dimension and $(x_1,\dots,x_n)$8 is (embedded) codimension; for a hypersurface this yields the single-factorial bound $(x_1,\dots,x_n)$9, greatly improving upon doubly-exponential or tower-exponential bounds of classical Hironaka-type algorithms (Brais, 1 Dec 2025).

Empirically, computational implementations (e.g., resweighted.lib in SINGULAR) report that for complicated singularities, the number of blow-ups required by the weighted scheme is dramatically fewer than for history-dependent or smooth-center schemes. For example, in $X = \Spec A$0 for the singularity $X = \Spec A$1,

• Weighted: $X = \Spec A$2 blow-ups, $X = \Spec A$3 charts
• Villamayor: $X = \Spec A$4 blow-ups, $X = \Spec A$5 charts

(Lee, 2020). This suggests that in practical computation, factorial reductions transfer to observable efficiency gains.

5. Extensions: Higher Codimension, Stacks, and Logarithmic Settings

For ideals of arbitrary codimension, the method proceeds via principalization in an ambient smooth variety: embed $X = \Spec A$6 into $X = \Spec A$7 of dimension $X = \Spec A$8, apply the weighted/log resolution, restrict back, and account for quotient singularities via stack-theoretic constructions. At each stage, weighted blow-ups are performed in the category of smooth log-DM stacks; exceptional divisors are simple normal crossings in the stack sense; the stacky nature does not impede iteration or functoriality, as stacky and coarse moduli modifications are connected in the final output (Brais, 1 Dec 2025, Quek, 2020, Temkin, 2023).

Logarithmic schemes and toroidal stacks enable these procedures to extend to contexts such as fs (fine-and-saturated) log schemes, toroidal Deligne–Mumford stacks, and even non-archimedean analytic spaces, provided the requisite functoriality (classically: under smooth maps, logarithmic: under log-smooth maps) (Temkin, 2023, Quek, 2020). The invariants, admissible centers, and transformations generalize to include Kummer covers, monomial ideals, and toroidal blow-ups, always preserving or restoring log smoothness and principalization properties (Abramovich et al., 17 Mar 2025, Temkin, 2023, Quek, 2020).

6. Worked Examples and Structural Properties

Several illustrative examples clarify the algorithmic process:

• For $X = \Spec A$9, the maximal order is $A$0, and one weighted blow-up along $A$1 produces a smooth chart (Lee, 2020).
• In the log/toroidal context, ideals like $A$2 on $A$3 are resolved by a single blow-up along $A$4, capitalizing on the log structure (Temkin, 2023, Abramovich et al., 17 Mar 2025).

Characteristic features of these methods include:

• Immediate improvement at every step: the maximal invariant strictly drops after each blow-up.
• Canonical (choice-free) selection of centers: centers are always at the maximal locus, and their structure is determined solely from the current singularity, not any historical data.
• Functorial behavior: algorithms and outputs depend only on the current scheme, invariant, and divisor/log-structure.
• Reduction to classical (Hironaka-type) resolution possible after destackification and toroidal normalization (Quek, 2020, Temkin, 2023).

7. Comparative and Contextual Analysis

The main distinctions between logarithmic, weighted, and classical resolution can be summarized as follows:

Algorithmic Class	Invariant Type	Centers	Functoriality	Complexity Bound
Classical	Pairs/Boundary	Smooth	Smooth maps	Doubly-exponential
Logarithmic	Log order	Log centers	Log-smooth maps	Improved, few steps
Weighted	Rational/lex tuple	Weighted, stacky	All smooth maps	Factorial $A$5

Classical resolution requires bookkeeping of exceptional divisors, monomial stages, and has non-canonical center selection. Logarithmic and weighted schemes eliminate history, unify all choices under a single invariant, and operate via canonical weighted blow-ups or Kummer blowings-up, effecting principalization in far fewer steps and opening paths to generalizations across broader mathematical contexts (Brais, 1 Dec 2025, Temkin, 2023, Quek, 2020).

Major open questions concern precise complexity estimates in relation to input degree and generalization to positive characteristic, where the stacky/weighted machinery does not directly apply due to wild ramification and failure of structure theorems (Abramovich et al., 17 Mar 2025, Temkin, 2023, Quek, 2020).

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