On minimal model program and Zariski decomposition of potential triples

Dae-Won Lee; Sung Rak Choi; Sungwook Jang

REVIEW 2 major objections 4 minor 16 references

On minimal model program and Zariski decomposition of potential triples

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A divisor with a birational Zariski decomposition converts a potential triple into a generalized pair, making the potentially non-klt locus closed and the $(K_X+\Delta+D)$-MMP runnable.

desk verdict A clean reduction from potential triples with birational Zariski decomposition to generalized pairs; the proof is sound modulo a standard fact that should be made explicit, and the main theorem deserves peer review. read the letter →

arxiv 2502.00790 v1 pith:JZREYXGM submitted 2025-02-02 math.AG

Sung Rak Choi , Sungwook Jang , Dae-Won Lee This is my paper · ORCID

classification math.AG MSC 14E3014J17

keywords potentialtriplegeneralizedpairZariskidecompositionminimalmodelprogramlogdiscrepancypkltplcasymptoticvaluation

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

Potential triples are a broad setting: a normal projective variety $X$, an effective divisor $\Delta$, and a pseudoeffective $\mathbb{R}$-Cartier divisor $D$ (a real divisor in the closure of the effective cone), with no nefness required of $D$. Singularities are measured by the potential log discrepancy $a(E;X,\Delta,D)=a(E;X,\Delta)-\sigma_E(D)$, where $\sigma_E(D)$ is the asymptotic valuation of $D$ along a prime divisor $E$. This paper proves that when $D$ has a birational Zariski decomposition $f^*D=P+N$, the triple behaves exactly like the generalized pair $(X,(\Delta+f_*N)+f_*P)$: pklt and plc triples become gklt and glc pairs, and the potentially non-klt locus $\mathrm{pNklt}(X,\Delta,D)$ equals the generalized non-klt locus, hence is Zariski closed. Consequently, for a $\mathbb{Q}$-factorial plc triple one can run the $(K_X+\Delta+D)$-MMP by importing the known generalized MMP, though termination is not claimed. As an application, a $\mathbb{Q}$-factorial pklt pair $(X,\Delta)$ for which $-(K_X+\Delta)$ admits a birational Zariski decomposition with NQC positive part has a $-(K_X+\Delta)$-minimal model.

What carries the argument

The load-bearing object is the birational Zariski decomposition $f^*D=P+N$, where $f:Y\to X$ is a projective birational morphism, $P$ is nef (or NQC in the application), and $N$ is effective with coefficients given by the asymptotic valuations of $D$. The mechanism is the equality $a(E;X,\Delta,D)=a(E;X,\Delta)-\mathrm{mult}_E(N)$ for every prime divisor $E$ on a common model, which rewrites the potential log discrepancy as the generalized log discrepancy of $(X,(\Delta+f_*N)+f_*P)$. This identification carries the whole argument: it turns the potential triple into a generalized pair, so the closedness of the potentially non-klt locus and the availability of the $(K_X+\Delta+D)$-MMP are inherited from the generalized-pair theory.

What would settle it

Compute, on a smooth projective threefold, a birational Zariski decomposition $f^*D=P+N$ and then take a prime divisor $E$ on a higher model $Y'\to Y$; if $\sigma_E(D)\neq\mathrm{mult}_E(N)$ for that $E$, the discrepancy equality in Theorem 3.1 fails and the reduction to generalized pairs collapses. A more global refutation would be a $\mathbb{Q}$-factorial plc triple satisfying the hypothesis whose $\mathrm{pNklt}$ locus is not Zariski closed; none is known, and the standard nonclosed diminished-base-locus example does not satisfy the hypothesis because it admits no birational Zariski decomposition.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that a potential triple $(X,\Delta,D)$ whose divisor $D$ admits a birational Zariski decomposition $f^*D=P+N$ has two properties: if it is plc, then $\mathrm{pNklt}(X,\Delta,D)$ is Zariski closed; and if it is $\mathbb{Q}$-factorial and plc, then the $(K_X+\Delta+D)$-MMP can be run. The proof rests on Theorem 3.1, which identifies the potential log discrepancy of any divisor $E$ over $X$ with the generalized log discrepancy of $(X,(\Delta+f_*N)+f_*P)$: writing $K_Y+\Delta_Y=f^*(K_X+\Delta)$, one gets $a(E;X,\Delta,D)=1-\mathrm{mult}_E(\Delta_Y+N)$, so the singularities of the triple and of the associated generalized pair coincide and $\mathrm{pNklt}(X,\Delta,D)=\mathrm{gNklt}(X,(\Delta+f_*N)+f_*P)$. Because the generalized non-klt locus is Zariski closed and the generalized MMP is available for glc pairs, the two conclusions follow. A corollary is the anticanonical statement: a $\mathbb{Q}$-factorial pklt pair $(X,\Delta)$ with $-(K_X+\Delta)$ admitting a birational Zariski decomposition whose positive part is NQC (a nonnegative $\mathbb{Q}$-linear combination of nef $\mathbb{Q}$-Cartier divisors) admits a $-(K_X+\Delta)$-minimal model.

Load-bearing premise

The load-bearing premise is that, after passing to a common higher birational model, the asymptotic valuation $\sigma_E(D)$ of $D$ along a prime divisor $E$ equals the coefficient of $E$ in the negative part $N$ of a birational Zariski decomposition $f^*D=P+N$; the paper uses this equality in Theorem 3.1 without proving it, and if it failed for some exceptional divisor the identification $\mathrm{pNklt}(X,\Delta,D)=\mathrm{gNklt}(X,(\Delta+f_*N)+f_*P)$ would break.

Editorial extensions

If this is right

For every $\mathbb{Q}$-factorial plc triple $(X,\Delta,D)$ with $D$ admitting a birational Zariski decomposition, a $(K_X+\Delta+D)$-MMP exists as a sequence of divisorial contractions and flips; termination is not guaranteed.
Under the same hypothesis, $\mathrm{pNklt}(X,\Delta,D)$ is Zariski closed for plc triples, matching $\mathrm{gNklt}$ of the associated generalized pair.
A $\mathbb{Q}$-factorial pklt pair $(X,\Delta)$ whose anticanonical divisor $-(K_X+\Delta)$ admits a birational Zariski decomposition with NQC positive part admits a $-(K_X+\Delta)$-minimal model.
If $D$ is a big $\mathbb{Q}$-Cartier divisor and no plc center of $(X,\Delta,D)$ lies in the augmented base locus $B_+(D)$, then $\mathrm{pNklt}(X,\Delta,D)=\mathrm{Nklt}(X,\Delta+D')$ for some $D'\sim_{\mathbb{Q}}D$, so $(X,\Delta+D')$ is lc and the $(K_X+\Delta+D)$-MMP can be run.
Potential log discrepancies are nondecreasing along $(K_X+\Delta+D)$-negative contractions, so every intermediate step of the MMP is again a plc triple.

Reading between the lines

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

The generalized-pair route only opens the MMP for the Zariski-decomposable case; the paper's own note that a Cone theorem and Contraction theorem for potential triples is future work indicates that a genuinely potential MMP would need new machinery, not just this translation.
One testable question the paper leaves open is whether the associated generalized pair depends on the choice of birational Zariski decomposition; if two choices produced different $\mathrm{gNklt}$ loci, the equality $\mathrm{pNklt}=\mathrm{gNklt}$ would still hold for each choice, but the comparison would be decomposition-dependent.
The NQC hypothesis in the anticanonical corollary may be an artifact of the cited generalized-pair minimal-model theorem; testing whether the minimal model exists without NQC would show whether potential pairs are actually more flexible than generalized pairs in this respect.

Share X Bluesky LinkedIn Reddit HN

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

Dear [Colleague],

The thing to know: this is a short, honest paper that gives a clean reduction. If D admits a birational Zariski decomposition, then a potential triple (X,Δ,D) is just a generalized pair in disguise, so the entire [CHLX] generalized MMP applies. Theorem 3.1 is the heart, and it is correct. The generality — arbitrary pseudoeffective D, not just −(K_X+Δ) or the special big cases in [CJ], [CJL], [Jan] — is genuinely new. The paper clearly says termination is not guaranteed and that the general case without Zariski decomposition remains open. That honesty is worth something.

The proof of Theorem 3.1 has a small gap that the stress-test caught. When the text says 'taking Y higher if necessary' and then uses σ_E(D)=mult_E(N), it is using the fact that pulling back a Zariski decomposition preserves the negative part: if P is nef then g*P is nef, and the standard Nσ property gives σ_E(g*f*D)=mult_E(g*N). That is a known fact (it is essentially Nakayama's definition), but the paper should state it or cite it explicitly rather than leaving it implicit in the word 'higher'. This is a minor fix, not a flaw in the argument.

The bigger soft spot is Theorem 1.3. It outsources the crucial existence of D' to [CJK, Lemma 4.6], an unpublished preprint with overlapping authors. The main theorem (1.1) doesn't rely on that, but the advertised application does. A referee should ask for a public proof or a published reference before Theorem 1.3 is used.

Corollary 1.2 is essentially Jang's theorem; the authors acknowledge it with 'cf.' and reprove it via [TX]. Fine, but don't buy the paper for that.

Overall: the central reduction is sound, the exposition is readable, and the paper knows what it doesn't know. This deserves a serious referee. My recommendation: send it to review, with a request for a short clarification of the pullback compatibility and for the [CJK] dependency to be resolved. A competent referee can fill the rest.

Best,

Referee Report

2 major / 4 minor

Summary. The paper studies potential triples (X,∆,D), where ∆ is a boundary divisor and D is a pseudoeffective R-Cartier divisor, and shows that when D admits a birational Zariski decomposition f*D = P + N, the potential triple can be encoded as a generalized pair (X,(∆+f_*N)+f_*P). The main result (Theorem 1.1) states that for such triples the potential non-klt locus pNklt(X,∆,D) is Zariski closed, and that in the Q-factorial plc case one can run the (K_X+∆+D)-MMP. The proof goes through a comparison theorem (Theorem 3.1) identifying potential log discrepancies with generalized log discrepancies, and then invokes the established MMP for generalized pairs [CHLX]. Applications include Corollary 1.2 on the existence of anticanonical minimal models when -(K_X+∆) admits a birational Zariski decomposition with NQC positive part, and Theorem 1.3 on effective representatives of a big Q-divisor D under an augmented-base-locus condition.

Significance. The central idea is attractive: it creates a direct bridge between potential triples and the now well-developed MMP for generalized pairs, yielding both a Zariski-closedness statement and a runnable MMP in a setting strictly larger than generalized pairs. The paper is clearly organized, and the reduction in Theorem 3.1, once its missing compatibility lemma is supplied, is conceptually clean. The examples in Example 2.7 usefully delineate the boundary of the hypothesis. If the proof gap identified below is repaired, the results would be a meaningful extension of [Jan] and a useful tool for studying anticanonical divisors with Zariski decompositions.

major comments (2)

[Section 3, Theorem 3.1 (proof)] The proof states 'By taking Y higher if necessary, we can assume that E is a divisor on Y' and then uses the equality a(E;X,∆,D)=1-mult_E(∆_Y+N). This assumes that for every higher birational model g:W→Y, the negative part of the divisorial Zariski decomposition of (fg)^*D is exactly g^*N, i.e. that σ_E(D)=mult_E(g^*N) for every prime divisor E on W. This pullback compatibility is not a formal consequence of the definition of a birational Zariski decomposition and is not proved or precisely referenced in the text. Without it, the equality between potential and generalized log discrepancies can fail on exceptional divisors of W over Y, and the identification pNklt(X,∆,D)=gNklt(X,(∆+f_*N)+f_*P) would not follow. The authors should state and prove a lemma (or cite the exact [Nak] statement, e.g. [Nak, Lemma III.5.15]) showing that if P is nef and N=Nσ(f*D), then Nσ(g*f*D)=g^*N for every higher birational morphism g.
[Section 3, Theorem 1.3 (proof)] The proof constructs an effective divisor D'∼_Q D with pNklt(X,∆,D)=Nklt(X,∆+D') and (X,∆+D') lc by citing '[CJK, Proof of Proposition 4.9]'. Since [CJK] is an unpublished preprint, this is an unverifiable dependency for a central theorem of the paper. The authors should either include a self-contained proof of the required statement or supply a published reference; otherwise Theorem 1.3 is not fully supported.

minor comments (4)

[Notation, throughout] The notation f*N in Theorem 3.1 and elsewhere is used for the pushforward f_*N, while f*D denotes the pullback; this is confusing. Please introduce a consistent notation such as f_*N for pushforwards.
[Throughout] The text contains several typographical and OCR errors, e.g. 'a nalyzing', 'p otential', 'the generalized pairs', and the arrow symbols rendered as '/axisshort/axisshort/arrowaxisright'. A careful proofreading pass is needed.
[Section 3, Proposition 3.5 (proof)] The proof asserts that for an ample divisor A on Y, B(f*D + A) ⊆ Supp(N) because B(P + A) = ∅; this uses the standard fact that for an ample divisor L and an effective divisor N, B(L+N) ⊆ Supp(N). A one-sentence justification would make the argument easier to follow.
[Section 2.5, Definition 2.4] The definition of a potential triple refers to 'a pair (X,∆)' without explicitly stating that ∆ is effective; since effectiveness of ∆ is used in the constructions of generalized pairs, it should be stated in the definition.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1.1 reduces to the external generalized-pair MMP of [CHLX], and the self-citations are not load-bearing.

full rationale

The central derivation is not circular. In Theorem 3.1, the potential log discrepancy a(E;X,Δ,D)=a(E;X,Δ)-σ_E(D) is equated with the generalized log discrepancy of (X,(Δ+f_*N)+f_*P) by substituting mult_E N for σ_E(D). For divisors E on the model Y carrying the birational Zariski decomposition, this is exactly the definition of N as the negative part, so the identification is definitionally tight, not an import of the target result. For divisors on higher models the proof says 'taking Y higher if necessary'; this is a genuine potential gap in justification and a correctness risk, but it is not circularity: it does not assume the conclusion, and it could be repaired by an external lemma from [Nak] if available. The MMP statement in Theorem 1.1(2) is obtained by the equality K_X+(Δ+f_*N)+f_*P = K_X+Δ+D and then imported from the generalized-pair MMP of [CHLX], whose authors do not overlap with the present paper, so the load-bearing step is independent external input rather than a self-citation. The self-citations [CJK], [CJL], [Jan], and [CJ] are contextual: for example, the proof of Corollary 1.2 uses [TX, Theorem 5.18] rather than [Jan], and [CJK, Lemma 4.6] is used only in the introductory heuristic discussion, not in the proof of Theorem 1.1. The paper itself explicitly flags that the general potential-triple MMP is not established and would require future Cone and Contraction theorems (Section 2.6), which further indicates that no conclusion is being smuggled in by assumption. Overall, no step in the derivation reduces by construction to its own input, so the circularity score is minimal.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants appear. The central construction is definitional: with f*D=P+N and P nef, the pair (X,(∆+f_*N)+f_*P) is a generalized pair. The analytic load is carried by the cited MMP theorems; the most fragile input is the unpublished, author-overlapping [CJK, Prop 4.9]. The auxiliary ε in Corollary 1.2 is a generic small constant, not a fitted parameter.

assumptions (5)

standard math MMP for Q-factorial glc generalized pairs (CHLX, Theorem 2.3)
Used as the engine to run the (K_X+∆+D)-MMP after the reduction in Theorem 3.3.
standard math Existence of minimal models for gklt pairs with NQC positive part (TX, Theorem 5.18)
Used in Corollary 1.2 to obtain the anticanonical minimal model; assumes the NQC condition on the positive part.
standard math Existence of D'∼Q D with (X,∆+D') lc and Nklt equality (CJK, Proposition 4.9)
Outsourced key step in Theorem 1.3; the preprint is by overlapping authors and the hypotheses are not restated.
standard math Asymptotic valuation and divisorial Zariski decomposition properties (Nakayama)
Underlies σ_E(D), the definition of pNklt, and the identification σ_E(D)=mult_E N used in Theorem 3.1.
domain assumption Generalized pair convention allowing M=f_*M_Y as an R-divisor with only K_X+B+M R-Cartier
The reduction sets M=f_*P, which may not be R-Cartier on X; the paper's Section 2.1 follows this convention and [CHLX] must share it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On minimal model program and Zariski decomposition of potential triples." pith.science (2026). https://pith.science/paper/JZREYXGM

@misc{pith2026250200790,
  author       = {Pith},
  title        = {Pith review of: On minimal model program and Zariski decomposition of potential triples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JZREYXGM}},
  note         = {Machine review of arXiv:2502.00790}
}

abstract

In this paper, we investigate properties of potential triples $(X,\Delta,D)$ which consists of a pair $(X,\Delta)$ and a pseudoeffective $\mathbb{R}$-Cartier divisor $D$. In particular, we show that if $D$ admits a birational Zariski decomposition, then one can associate a generalized pair structure to the potential triple $(X,\Delta,D)$. Moreover, we can run the generalized MMP on $(K_X+\Delta+D)$ as special cases. As an application, we also show that for a pklt pair $(X,\Delta)$, if $-(K_X+\Delta)$ admits a birational Zariski decomposition with $\mathrm{NQC}$ positive part, then there exists a $-(K_X+\Delta)$-minimal model.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 13 canonical work pages

[1]

Birkar and Z

C. Birkar and Z. Hu. Log canonical pairs with good augmented base loci, Compos. Math., 150, 579--592 (2014)

work page 2014
[2]

Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math

C. Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J., 136(1), 173--180 (2007)

work page 2007
[3]

Cacciola and L

S. Cacciola and L. Di Biagio, Asymptotic base loci on singular varieties, Math. Z., 275(1--2), 151--166 (2013)

work page 2013
[4]

G. Chen, J. Han, J. Liu, and L. Xie, Minimal model program for algebraically integrable foliations and generalized pairs, arXiv:2309.15823v2 https://arxiv.org/abs/2309.15823, (2023)

arXiv 2023
[5]

Choi and S

S. Choi and S. Jang, ACC of PLC threshold, Manuscripta Math., 174, 491--503 (2024)

work page 2024
[6]

S. Choi, S. Jang, and D. Kim, Adjoint asymptotic multiplier ideal sheaves, arXiv:2311.07441 https://arxiv.org/abs/2311.07441, (2024)

arXiv 2024
[7]

S. Choi, S. Jang, and D.-W. Lee, Plc pairs with good asymptotic base loci, arXiv:2411.04628 https://arxiv.org/abs/2411.04628, (2024)

arXiv 2024
[8]

Choi and J

S. Choi and J. Park, Potentially non-klt locus and its applications, Math. Ann., 366(1), 141--166 (2016)

work page 2016

Show all 16 references

[9]

G. D. Chen and N. Tsakanikas. On the termination of flips for log canonical generalized pairs, Acta Math. Sin. Engl. Ser., 39(6), 967--994 (2023)

2023
[10]

L. Ein, R. Lazarsfeld, M. Musta t a , M. Nakamaye, and M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), 56(6), 1701--1734 (2006)

2006
[11]

C. D. Hacon and J. Liu, Existence of flips for generalized lc pairs, Camb. J. Math., 11(4), 795--828 (2023)

2023
[12]

Jang, Anticanonical minimal models and Zariski decomposition, arXiv:2405.10533 https://arxiv.org/abs/2405.10533, (2024)

S. Jang, Anticanonical minimal models and Zariski decomposition, arXiv:2405.10533 https://arxiv.org/abs/2405.10533, (2024)

2024 arXiv
[13]

Lesieutre, The diminished base locus is not always closed, Compos

J. Lesieutre, The diminished base locus is not always closed, Compos. Math., 150(10), 1729--1741 (2014)

2014
[14]

Nakayama, (2004)

N. Nakayama, (2004). Zariski-decomposition and abundance (Vol. 14). Tokyo: Mathematical Society of Japan

2004
[15]

Tsakanikas and L

N. Tsakanikas and L. Xie, Remarks on the existence of minimal models of log canonical generalized pairs, Math. Z., 307(1), 1--39 (2024)

2024
[16]

Tsakanikas and Z

N. Tsakanikas and Z. Xie, Comparison and uniruledness of asymptotic base loci, https://arxiv.org/abs/2309.01031 arXiv:2309.01031v2 , (2024)

2024 arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Birkar and Z

C. Birkar and Z. Hu. Log canonical pairs with good augmented base loci, Compos. Math., 150, 579--592 (2014)

work page 2014

[2] [2]

Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math

C. Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J., 136(1), 173--180 (2007)

work page 2007

[3] [3]

Cacciola and L

S. Cacciola and L. Di Biagio, Asymptotic base loci on singular varieties, Math. Z., 275(1--2), 151--166 (2013)

work page 2013

[4] [4]

G. Chen, J. Han, J. Liu, and L. Xie, Minimal model program for algebraically integrable foliations and generalized pairs, arXiv:2309.15823v2 https://arxiv.org/abs/2309.15823, (2023)

arXiv 2023

[5] [5]

Choi and S

S. Choi and S. Jang, ACC of PLC threshold, Manuscripta Math., 174, 491--503 (2024)

work page 2024

[6] [6]

S. Choi, S. Jang, and D. Kim, Adjoint asymptotic multiplier ideal sheaves, arXiv:2311.07441 https://arxiv.org/abs/2311.07441, (2024)

arXiv 2024

[7] [7]

S. Choi, S. Jang, and D.-W. Lee, Plc pairs with good asymptotic base loci, arXiv:2411.04628 https://arxiv.org/abs/2411.04628, (2024)

arXiv 2024

[8] [8]

Choi and J

S. Choi and J. Park, Potentially non-klt locus and its applications, Math. Ann., 366(1), 141--166 (2016)

work page 2016