Log Resolution of Singularities

• Log resolution of singularities is a framework that uses controlled blow-ups to resolve both a variety and its boundary into a simple normal crossings structure.
• It employs effective methods in characteristic zero with primitive-recursive bounds, using invariants like log-order and weighted or stack-theoretic techniques.
• These techniques underpin advances in birational geometry, mirror symmetry, and motivic invariants, and are implemented via explicit, finite-time algorithms.

A log resolution of singularities is a central operation in birational and logarithmic algebraic geometry, providing a framework to resolve singularities of a variety in a manner compatible with a prescribed divisor—typically, one with simple normal crossings (snc) structure. Log resolutions refine classical resolution of singularities by encoding boundary data and are key to developments in birational geometry, minimal model programs, mirror symmetry, and the theory of motivic invariants. In characteristic zero, recent advances have delivered both effective primitive-recursive bounds for ambient dimension and degrees in explicit log resolutions (Bierstone et al., 10 Nov 2025), and functorial, stack-theoretic, or weighted approaches with algorithmically controlled invariants (Temkin, 2023, Abramovich et al., 17 Mar 2025, Corti et al., 14 Mar 2025, Satriano et al., 2023).

1. Definitions and Core Notions

Let $X\subset \mathbb{P}^n$ be a projective variety over an algebraically closed field of characteristic zero, and $E \subset \mathbb{P}^n$ a reduced simple normal crossings divisor (snc). The divisor $E$ is snc at $p\in\mathbb{P}^n$ if, locally at $p$, there exist regular coordinates such that $E$ is defined by $x_{j_1}\cdots x_{j_r}=0$ for some $r$ and distinct coordinate indices. A log resolution of the pair $(X,E)$ is a projective birational morphism

$\varphi: (X',E') \longrightarrow (X,E)$

obtained through a finite composition of blow-ups along smooth centers making normal crossings with existing exceptional divisors, ensuring:

• $X'$ is smooth,
• $$E' = \varphi^{-1}_\ast E + \operatorname{Exc}(\varphi)$$ is an snc divisor on $X'$,
• $\varphi$ is an isomorphism over $X \setminus \operatorname{Sing}(X \cup E)$.

Log resolutions simultaneously resolve $X$ and principalize the boundary $E$, facilitating invariants' calculations and canonical bundle control.

2. Effective Log Resolution in Characteristic Zero

The Bierstone–Grigoriev–Milman–Włodarczyk theorem provides primitive-recursive, effective bounds in the projective setup (Bierstone et al., 10 Nov 2025). For $X\subset\mathbb{P}^n$, $E\subset\mathbb{P}^n$ with $\deg X, \deg E \leq d$, there is an explicit, primitive-recursive function $F(n,d)=(n',d')$ such that there exists a log resolution

$\varphi:(X',E') \to (X,E)$

with $X'\subset\mathbb{P}^{n'}$, $\deg X', \deg E' \leq d'$, and with $(n',d')$ computable by closed-form recursion on $(n,d)$. The growth of $n'$ and $d'$ is controlled as follows:

• $M(n,d) \approx d^{n^2}$,
• $R(n,d) = (M(n,d)+1)!$ (in the primitive-recursive class $\mathcal{E}_3$),
• After $R(n,d)$ blow-ups, the dimension satisfies $n' \leq 2^{R(n,d)} n$,
• Final degrees $d' \leq (2d)^{2^{n'+2}}$.

Key lemmas ensure that, at each step, degrees of new equations are bounded by double exponentials in $n$. The entire construction is algorithmic: monomial–residual ideal factorization, construction and resolution of companion ideals, reduction to maximal order via passage to coefficient ideals on hypersurfaces of maximal contact, and explicit control of invariants ensure finite, effective termination. All steps, including derivation, coefficient ideal generation, and center choice, are constructive.

3. Logarithmic and Weighted Resolution Algorithms

Logarithmic and weighted methods extend and systematize effective log resolution, introducing new functorial invariants and flexibility, especially for log schemes, stacks, and pairs (Temkin, 2023, Abramovich et al., 17 Mar 2025). The key points are as follows:

• Log schemes: Schemes equipped with a sheaf of monoids tracking boundary information; log-smoothness generalizes toroidal/snc structures.
• Resolution process: Iterative Kummer (stacky) or weighted blow-ups along admissible centers (combinations of monomial and regular loci, possibly with rational exponents and stack structure).
• Invariants: The central resolution invariant is typically the maximal log-order (logarithmic order computed with log-derivations). Weighted methods assign to each ideal an upper-semicontinuous well-ordered tuple, and centers are constructed as loci where invariants are maximal.
• ** termination**: At each algorithmic step, blowing up along the current invariant's maximal locus reduces the invariant strictly lexicographically, ensuring finiteness.

Weighted blow-up constructions (Abramovich et al., 17 Mar 2025) involve defining, for an ideal $J=(x_1^{a_1},\ldots,x_k^{a_k})$ (with exponents $a_i=\ell/w_i$), an associated graded algebra and forming stacky quotients, with exceptional divisors that remain snc and with functorial behavior under base change.

4. Variants: Relative, Stacky, and Log Crepant Resolutions

Log resolutions generalize in several key directions:

• Relative log resolutions: Consider morphisms $f:X \to B$ with $B$ log-regular. Relative log-derivations and relative log-order allow principalization functorial for base changes, supporting semistable reduction for morphisms (Temkin, 2023).
• Stacky/Artin stack resolutions: For $X$ with log-terminal singularities, crepant resolutions by smooth Artin stacks always exist (Satriano et al., 2023). The construction proceeds by strong resolution, root stack modifications to adjust discrepancies, and stack-theoretic fiber products tying in moduli of degenerations.
• Log crepant log resolutions: For specific classes of singular log schemes (3-folds with prescribed combinatorial data), there is a conjectural existence of projective, log-smooth, log-crepant resolutions, motivated both by explicit weighted/toroidal blow-up constructions and by mirror symmetry via the Gross–Siebert program (Corti et al., 14 Mar 2025).

Strategies for log (or log crepant) resolution in these settings include local toroidal/weighted blow-ups, global smoothing families, scattering diagram constructions, and induction/mutation among combinatorial data.

5. Examples and Algorithmic Schemes

Worked examples include:

• Simple log-case with Kummer center: $X= \mathrm{Spec} \, k[t,u]$, $I=(t^2+u)$; log-order at $(0,0)$ is 2; Kummer blow-up (after adjoining $u^{1/2}$) at $(t, u^{1/2})$ produces a log-smooth model (Temkin, 2023).
• Weighted blow-up for the plane cusp: $X \subset \mathbb{A}^2_{x,y}$ defined by $y^2-x^3=0$; a single weighted blow-up with center $J=(x^3, y^2)$ and weights $(2,3)$ drastically lowers the invariant, further reducing to log-smoothness after two further ordinary blow-ups (Abramovich et al., 17 Mar 2025).
• A$_n$-singularity and Tom–Jerry log data: Explicit small/weighted blow-ups with controlled centers and local equations produce crepant, log-smooth (besides specified orbifold points) projective resolutions, matching the combinatorial kink data (Corti et al., 14 Mar 2025).

These examples demonstrate how the resolution invariants and blow-up algorithms, whether ordinary, stacky, or weighted, reduce complex singularities to snc structure in a controlled, often combinatorial, manner.

6. Connections to Mirror Symmetry and Advanced Research Directions

Log resolution and log crepant techniques underpin major developments beyond classical desingularization. In mirror symmetry and the Gross–Siebert program, initial wall data (zero-mutable combinatorial structures) encode both the smoothing and the canonical theta functions (tropical mirror construction). The conjecture that every zero-mutable log datum admits a canonical, distinguished log crepant resolution, and that its smoothing yields the genuine smoothing component of a toric 3-fold, ties together combinatorics, tropical geometry, and birational (log/minimal model) geometry (Corti et al., 14 Mar 2025).

Moreover, the existence of stack-theoretic crepant resolutions for all log-terminal singularities extends the range of the stringy McKay correspondence, providing new avenues for the paper of motivic invariants and stringy Hodge theory (Satriano et al., 2023). Motivic integration and change-of-variables formulas can be carried out in Artin-stack settings, ensuring equality of stringy and classical invariants when crepant stacky resolutions exist.

7. Algorithmic and Computational Aspects

All of the effective, weighted, and stacky log resolution methods in characteristic zero are constructive: each step (factorization into monomial/residual ideals, construction of coefficient ideals, selection of weighted/Kummer centers, computation of transforms and invariants) can be implemented in a computer algebra system, with explicit, finite-time algorithms and degree/dimension bounds. The lexicographic drop of the invariant at each blow-up, together with double-exponential degree bounds per step, ensures primitive-recursive global bounds on complexity.

Noetherian induction is avoided in favor of strictly decreasing well-ordered invariants, and all essential data (dimensions, degrees, exceptional divisors, and transformed equations) are explicitly controlled. These guarantees make log resolution, as currently developed, not merely a theoretical but also an algorithmically accessible tool for modern algebraic geometry in characteristic zero.