Minimal log discrepancies on a fixed threefold

Masayuki Kawakita

REVIEW 3 major objections 4 minor 28 references

Minimal log discrepancies on a fixed threefold

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Minimal log discrepancies on any fixed normal threefold satisfy the ascending chain condition.

desk verdict Kawakita proves the fixed-variety ACC for minimal log discrepancies on threefolds, a major result, but a key step in the termination argument contains a false factoriality claim that needs repair. read the letter →

arxiv 2412.03006 v1 pith:IPR4IO4P submitted 2024-12-04 math.AG

Masayuki Kawakita This is my paper · ORCID

classification math.AG MSC 14E3014B0514J30

keywords minimallogdiscrepancyACCDCCthreefoldcanonicalpairsdivisorialcontractionsgenericlimitsbirationalgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that on any fixed normal threefold, the minimal log discrepancies obtained from boundaries with coefficients in a DCC set satisfy the ascending chain condition: no infinite strictly increasing sequence of such values exists. This is the fixed-variety case of a central conjecture in birational geometry, and it was previously known for smooth threefolds but open for singular ones. The author reduces the statement to a uniform boundedness assertion on a fixed terminal threefold germ, then builds an infinite sequence of divisorial contractions from a log canonical limit pair with a one-dimensional centre and shows that an integer invariant attached to each germ strictly decreases until the germ becomes smooth. If the proof is correct, the same mechanism also yields the ACC for a-lc thresholds, a uniform m-adic semi-continuity statement, and a boundedness statement for divisors computing minimal log discrepancies on the fixed threefold.

What carries the argument

Three objects carry the argument. First, the generic limit: an infinite sequence of $\mathbb{Q}$-divisors $\Delta_i$ on the germ is replaced by one boundary $\hat{\Delta}$ on the completion $\hat{X}$ of the local ring over a large algebraically closed field, so that the ACC problem becomes a statement about a single pair $(\hat{X},\hat{\Delta})$. Second, a birational model: from a log canonical pair whose smallest lc centre is a regular curve $\hat{C}$, the author constructs a contraction $Y_K\to X_K$ and, by Proposition 5.2, an infinite sequence of threefold divisorial contractions $E_{i+1}\subset X_{i+1}\to x_i\in X_i$ contracting a divisor to the point $x_i$ on the strict transform of $\hat{C}$. Third, two integer invariants at the Gorenstein indices of the sequence: $m_i$, the least order of the maximal ideal of $x_i$ along the future exceptional divisors, and $l_i$, defined as the length of the cokernel of the natural map $\bigwedge^2 I_i/I_i^{(2)}\to \omega_{X_i}\otimes \omega_{C_i}^{-1}$ for the ideal sheaf $I_i$ of the curve $C_i$. The classification of threefold divisorial contractions (Table 1) is used to show that, after $m_i$ is constant, $l_i$ strictly decreases until $x_i$ is smooth; this termination is Theorem 5.3, and it is the step from which the uniform bound $l$ is derived.

What would settle it

A concrete disproof would be a fixed normal threefold $X$ and a DCC set $I$ for which the values $\mathrm{mld}_x(X,\Delta_i)$ at a closed point form an infinite strictly increasing sequence, or, failing that, an infinite sequence of divisorial contractions from Proposition 5.2 with constant $m_i$ and non-decreasing $l_i$ at singular Gorenstein points, which would undermine Theorem 5.3 and hence the bound $l$.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: fix a normal threefold $X$ and a subset $I$ of the positive real numbers satisfying the DCC. Then the set $$\{\mathrm{mld}_\eta(X,\$\Delta$) \mid \eta \text{ scheme-theoretic point},\ (X,\$\Delta$)\text{ pair},\ \$\Delta$\in I\}$$ satisfies the ACC. Since every minimal log discrepancy at a scheme-theoretic point with closure $Z$ equals the minimal log discrepancy at a general closed point of $Z$ minus $\dim Z$, the closed-point case is the essential one. The proof of the theorem passes through Theorem 3.1, a boundedness statement on the germ of a $\mathbb{Q}$-factorial terminal threefold: for every sequence of effective $\mathbb{Q}$-divisors with denominators bounded by a fixed integer, there is a uniform bound $l$ such that infinitely many of the mld values are computed by divisors whose own log discrepancy with respect to the threefold is at most $l$.

Load-bearing premise

The argument relies on a complete numerical classification of threefold divisorial contractions; if a contraction of a terminal threefold not listed in Table 1 exists, the descent of the invariant that forces a smooth point could fail, and the bound proving the ACC would not follow.

Editorial extensions

If this is right

For every fixed normal threefold and every DCC coefficient set, the non-negative minimal log discrepancies at its scheme-theoretic points form an ACC set, so no infinite strictly increasing sequence of values occurs.
The ACC for a-lc thresholds follows (Theorem 1.2): for fixed $a$, the set of thresholds $t$ with $\mathrm{mld}_\eta(X,\Delta+tA)=a$ satisfies the ACC.
Uniform $m$-adic semi-continuity follows (Theorem 1.3): there is an integer $l$ such that ideals sharing their $l$-th powers modulo the maximal ideal give equal minimal log discrepancies.
A boundedness statement for computing divisors follows (Theorem 1.4): on a fixed log terminal threefold, every boundary's minimal log discrepancy is computed by a divisor whose own log discrepancy is bounded by a fixed $l$.
Together with the previously settled smooth case, the theorem completes the ACC for minimal log discrepancies on every fixed threefold.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's termination step suggests a prototype for higher dimensions: a fixed-variety ACC would follow from a decreasing, integer-valued statistic attached to the minimal centre along a sequence of divisorial contractions, and the pair $(m_i,l_i)$ shows what such a statistic can look like in dimension three.
Because the only classification input is the table of threefold divisorial contractions, a classification-free version of the descent lemma would likely make the uniform bound $l$ effective; one could compute it on explicit terminal quotient germs and compare the rate of descent of $l_i$.
A step the paper does not take is to let the threefold vary in a bounded family and ask whether the bound $l$ in Theorem 3.1 depends only on the family; a positive answer would reduce the unfixed three-dimensional ACC to a stratification of threefold germs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Kawakita proves the ACC for minimal log discrepancies on an arbitrary fixed normal threefold, extending his earlier smooth-threefold theorem. That is a substantial result: combined with lower semicontinuity it gives termination of flips for pairs on a fixed threefold, and via his previous equivalence it yields the boundedness statements in Theorems 1.2–1.4. The architecture is sensible: reduce to a fixed Q-factorial terminal germ, pass to a generic limit, construct models by divisorial contractions, and then use the numerical classification of threefold divisorial contractions to force a smooth model. The reliance on the author's own classification work is legitimate; those are published results and the paper is transparent about quoting them.

The proof as written has real presentation gaps. Theorem 4.4 is explicitly a reformulation of [13, Theorem 4.10] with the proof left to the reader; Lemma 7.10 ends with 'by a direct computation' for the final case. Those are not fatal, but they make independent verification slow.

The specific stress-test point about Lemma 7.4 is on target. The text says 'Xj is factorial at xj by Lemma 6.2.' That is false: Lemma 6.2 gives that every Q-Cartier divisor is linearly equivalent to a multiple of K_X, which, when xj is Gorenstein, makes every Q-Cartier divisor Cartier—but it does not make the local ring factorial. The ordinary double point is a terminal Gorenstein singularity that is not factorial. So the sentence as written is wrong. However, the conclusion the proof needs is not full factoriality but uniqueness of the π_ij-exceptional prime divisor through xj. That does follow from the non-containment assumption m_i O_Xj ⊄ m_j^2, using the facts that Xj is Q-factorial, xj is Gorenstein, and hence every exceptional divisor through xj is Cartier; two such divisors would put their product in m_j^2. The author doesn't supply that argument, so the proof has a genuine missing step, but it is repairable in a few lines.

I did not find circularity or invented entities. The citation pattern is heavy on self-citations, but they are to prior published classifications and the author's own verified results, which is appropriate here.

This is a paper for specialists in birational geometry and singularities. It deserves serious refereeing; a referee should ask the author to fix Lemma 7.4 and to expand the delegated proofs at least to the level of clear statements of the reductions.

Referee Report

3 major / 4 minor

Summary. The paper proves Theorem 1.1: for a fixed normal threefold X over an algebraically closed field of characteristic zero and a DCC set I of positive real numbers, the set of minimal log discrepancies mld_η(X,Δ) over all scheme-theoretic points η and boundaries Δ with coefficients in I satisfies the ACC. The proof reduces to a boundedness statement on a fixed Q-factorial terminal threefold germ (Theorem 3.1), constructs a generic limit pair (Rhat X,Rhat Δ), builds birational models by means of divisorial contractions (Propositions 5.1 and 5.2), and proves termination of the resulting sequence of contractions through two invariants m_i and l_i (Theorem 5.3). Theorems 1.2–1.4 are derived from Theorem 3.1 via equivalences established in the author's earlier work.

Significance. If the proof is correct, this is a substantial result: it establishes the ACC for minimal log discrepancies on every fixed threefold, including singular threefolds, and with it the equivalent uniform m-adic semi-continuity, ACC for a-lc thresholds, and Nakamura boundedness on a fixed threefold. The strategy is coherent and builds naturally on the author's prior classification of threefold divisorial contractions and on the smooth-threefold case. The paper also benefits from a clear reduction to a terminal germ and from the use of generic limits. However, the proof is long and relies heavily on delegated classifications and computations; the points raised below need to be addressed before the central claim can be considered verified.

major comments (3)

[§7, Lemma 7.4] The proof contains the sentence “Recall that X_j is factorial at x_j by Lemma 6.2.” This is not correct: Lemma 6.2 only states that every Q-Cartier divisor on a terminal threefold germ is linearly equivalent to an integer multiple of K_X. It does not imply factoriality of the local ring; a terminal Gorenstein ordinary double point is a counterexample. The subsequent conclusion that E_j is the only π_ij-exceptional prime divisor through x_j and that it is cut out by z_4 requires a proof. Since Lemma 7.4 is the engine that produces the strict descent of l_i in Propositions 7.9 and 7.11, and hence the termination argument for Theorem 5.3, this is a load-bearing gap. Please either prove the required factoriality (or the weaker Cartier/uniqueness statement) from the actual hypotheses, or replace the argument.
[§4, Theorem 4.4] The proof of Theorem 4.4 is delegated: the text says “The proof is the same as that of [13, theorem 4.10] ... one can write down the complete proof following the proof in [13] verbatim.” This theorem supplies the key equality mld_x(X,Δ_i) = mld_{y_i}(Y_i,Γ_i) or mld_{Rhat x}(Rhat X,Rhat Δ), so it is central to the reduction. The author explicitly notes that Rhat X may fail to be Q-factorial, which is a real difference from the smooth case. Please include a complete proof or a detailed lemma-by-lemma translation that makes the adaptation visible and verifiable, rather than leaving the proof to the reader.
[§7, Lemma 7.10] In the final case of Lemma 7.10, the assertion “Now by a direct computation of the weighted blow-up π_{i+1}, one can check that X_{i+2} has only quotient singularities” is not shown. This computation is used in Proposition 7.11(iii) to dispose of the case (e13, 2, 3) with a_{j+1}/n_{j+1} = 3/2. Please provide the computation in detail, or at least a precise reference to the classification result that contains it.

minor comments (4)

[§4, Lemma 4.1] The symbol l is used both as the index of the family approximation and as a rational bound in the conclusion; this overloading makes the statement harder to read and should be fixed.
[§4, proof of Theorem 4.4] The phrase “For the counterpart of the divisor Rhat Q” introduces an object Rhat Q that has not been defined in the proof; the notation should be introduced or the passage reworded.
[§6, Lemma 6.5(iii)] The terms B_ι(4) and B_ι(6) in the formula for d(2) are not defined in the paper; they should be defined or the reader should be pointed to the exact formula in [12].
[§7, proof of Theorem 5.3] The proof treats quotient singularities via Kawamata's result [15] without stating the precise statement used; a short statement of the needed assertion would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained relative to independently published prior results; the only flagged concern is a possible proof gap, not a circular dependence.

full rationale

The paper proves the ACC for minimal log discrepancies on a fixed threefold by reducing to Theorem 3.1, then to the generic limit and termination constructions of Sections 4-7. The reduction in Lemma 3.4 invokes the equivalence of ACC for minimal log discrepancies, ACC for a-lc thresholds, uniform m-adic semicontinuity, and Nakamura boundedness from the author's prior papers [11,13]; these are published results for smooth threefolds and fixed germs, not restatements of Theorem 1.1. The generic limit construction is taken from [11] and [18], the MMP and negativity lemma from [27] and [17], the ACC for canonical thresholds from [5], and the classification of threefold divisorial contractions from the author's prior classification papers [6]-[9] and the book [12]. These are independent published results with stated assumptions that do not include the target ACC. Theorem 5.3 and the invariant descent in Propositions 7.9 and 7.11 are genuine derivations using those classifications. The skeptic's concern about Lemma 7.4 is that Lemma 6.2 (Q-Cartier divisors are linearly equivalent to integer multiples of K_X) does not by itself imply factoriality; this is a possible correctness gap in the proof of strict descent of l_i, not a circular step. There is no fitted parameter renamed as a prediction, no self-citation chain that forces the conclusion by definition, and no renaming of a known result as a new organization. Hence the paper exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The proof does not introduce fitted constants or new entities. It relies on a substantial body of external results: the MMP for threefolds, the classification of threefold divisorial contractions, the ACC for canonical thresholds, the generic limit construction, and several equivalence theorems from the author's prior papers. None of these references assumes the central result. The new invariants mi and li are variations of known invariants, with li originally due to Mori, and are defined within the proof.

assumptions (8)

domain assumption The ground field is an algebraically closed field of characteristic zero.
Stated in Section 2; the MMP and classification results used throughout require characteristic zero.
standard math Existence of Q-factorial terminalisations and running of the MMP for threefolds.
Invoked in Lemma 3.2, Proposition 5.1, and the proof of Theorem 5.3; relies on [27] and standard threefold MMP results.
standard math Numerical and geometric classification of threefold divisorial contractions (Table 1, Theorems 6.3, 6.7, 6.9-6.11).
Central to the Section 7 termination proof of Theorem 5.3; cited to the author's papers [6]-[9] and book [12].
standard math ACC for canonical thresholds of terminal threefolds (Han-Liu-Luo [5, theorem 1.7]).
Used in Proposition 5.1 to prove the sequence of canonical thresholds terminates. External result, not proved in this paper.
standard math Generic limit construction for functions and divisors (Kollár [18]; Kawakita [11]).
Section 4 uses this construction to pass from an infinite sequence to a pair on a formal germ; properties are cited from [11] and [18].
standard math Boundedness of the Cartier index on klt varieties (Greb-Kebekus-Peternell [3, remark 1.11]).
Used in Lemma 3.4 to ensure coefficient sets remain DCC after passing to a terminalisation.
standard math Equivalence among ACC for mlds, a-lc thresholds, uniform m-adic semicontinuity, and Nakamura's boundedness on a fixed germ (Kawakita [11], [13]).
Lemma 3.4 uses these equivalences to derive Theorems 1.1-1.4 from Theorem 3.1.
standard math Q-factorial terminalisation of klt pairs over formal or analytic bases (Lyu-Murayama [21]).
Used in Lemma 4.3 to extract a divisor computing the mld at the generic point of the smallest lc centre.

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Pith. "Pith review of Minimal log discrepancies on a fixed threefold." pith.science (2026). https://pith.science/paper/IPR4IO4P

@misc{pith2026241203006,
  author       = {Pith},
  title        = {Pith review of: Minimal log discrepancies on a fixed threefold},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPR4IO4P}},
  note         = {Machine review of arXiv:2412.03006}
}

read the original abstract

We prove the ACC for minimal log discrepancies on an arbitrary fixed threefold.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 25 canonical work pages

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [13]

Minimal log discrepancies on smooth threefolds

M. Kawakita. Minimal log discrepancies on smooth three folds. arXiv:2312.13599

work page Pith review arXiv

[2] [1]

de Fernex, L

T. de Fernex, L. Ein and M. Mustat ¸˘ a. Shokurov’s ACC conj ecture for log canonical thresholds on smooth varieties. Duke Math. J. 152 (2010), 93–114

work page 2010

[3] [2]

de Fernex, L

T. de Fernex, L. Ein and M. Mustat ¸˘ a. Log canonical thres holds on varieties with bounded singularities. Classiﬁcation of algebraic varieties , 221–257. EMS Ser. Congr. Rep., European Mathematical Society, 2011

work page 2011

[4] [3]

D. Greb, S. Kebekus and T. Peternell. ´Etale fundamental groups of Kawamata log terminal spaces, ﬂat sheaves, and quotients of abelian varieties. Du ke Math. J. 165 (2016), 1965–2004

work page 2016

[5] [4]

Han and J

J. Han and J. Liu. On termination of ﬂips and exceptionall y non-canonical singularities. To appear in Geom. Topol

work page

[6] [5]

J. Han, J. Liu and Y. Luo. ACC for minimal log discrepancie s of terminal threefolds. arXiv:2202.05287

arXiv

[7] [6]

Kawakita

M. Kawakita. Divisorial contractions in dimension thre e which contract divisors to smooth points. Invent. Math. 145 (2001), 105–119

work page 2001

[8] [7]

Kawakita

M. Kawakita. Divisorial contractions in dimension thre e which contract divisors to compound A1 points. Compos. Math. 133 (2002), 95–116

work page 2002