Pith. sign in

REVIEW 2 major objections 4 minor 43 references

$P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper proves that, for generalised pairs whose auxiliary data is merely a nef R-divisor, the existence of a birational Nakayama–Zariski decomposition with nef positive part together with a weak nonvanishing condition (either the…

desk verdict The P-trivial MMP is a genuinely new idea and the 3-fold results are significant, but the construction's reliance on BCHM10 for non-NQC g-pairs is a load-bearing gap that is not justified in the paper. read the letter →

arxiv 2501.07551 v1 pith:D5UWX77O submitted 2025-01-13 math.AG

Zhengyu Hu This is my paper · ORCID
classification math.AG MSC 14E3014J17
keywords generalisedpairsminimalmodelprogramP-trivialMMPNakayama-Zariskidecompositionnon-NQCnefdivisorsspecialterminationlogabundancebirationalgeometry
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 new existence theorems for minimal models of generalised pairs, objects made of a variety, a boundary divisor, and a nef 'data' divisor. The central result is that if the log canonical divisor birationally has a Nakayama-Zariski decomposition with nef positive part, and either that positive part or the data divisor is numerically equivalent to an effective divisor, then a minimal model exists. This is proved for generalised klt pairs in all dimensions (Theorem 1.4) and for generalised lc pairs in dimension 3 (Theorem 1.5), with a dimension-4 result under a log-abundance hypothesis (Theorem 1.6). The main mechanism is the P-trivial MMP, whose steps preserve the nefness of the chosen nef divisor P, and which under these weak nonvanishing assumptions degenerates to a standard MMP and terminates. The results extend earlier work that required the data divisor to be a rational combination of nef divisors ('NQC') to the genuinely irrational, non-NQC case.

What carries the argument

The central object is the P-trivial MMP (Definition 4.5): an MMP with scaling of a b-ample divisor A in which each step is an MMP on $K_{X_i}+(B_i+\lambda_i A_i+M_i)+\alpha_i P_i$ with $\alpha_i\gg 0$ chosen so that the contracted extremal curve $C_i$ satisfies $P_i\cdot C_i=0$, hence nefness of P is preserved across the whole sequence. The supporting machinery comprises the boundedness of intersection numbers for extremal curves of minimal length (Lemma 3.8), a special termination theorem that lifts the MMP to small Q-factorialisations of g-lc centres (Theorem 4.19), and the degeneration criterion (Theorem 4.22) that under $P\equiv D\ge 0$ or $M\equiv D\ge 0$ the P-trivial MMP is actually a standard MMP on $K_X+B+M+\alpha P$; Lemma 4.30 then turns degeneration into termination in a minimal model.

What would settle it

Exhibit a projective 3-dimensional generalised lc pair $(X,B+M)$ with data $M'$ a nef $\mathbb{R}$-divisor such that $K_X+B+M$ birationally has a Nakayama-Zariski decomposition with nef positive part and either $K_X+B+M\equiv D\ge 0$ or $M\equiv D\ge 0$, yet the P-trivial MMP of Definition 4.5 produces an infinite sequence of flips that never reaches a P-minimal model. Concretely, check whether the auxiliary MMP on $K_X+(B+\lambda A+M)+\alpha P$ satisfies the klt hypotheses of [BCHM10] when $M'$ has irrational coefficients; a failure of termination there would give a counterexample to Theorem 1.5.

Watch

Extended reading notes

Core claim

The author develops a theory of P-trivial MMP: for a Q-factorial generalised dlt pair and a nef divisor P, one runs an MMP on $K_X+(B+\lambda A+M)+\alpha P$ with scaling of a b-ample divisor A, choosing $\alpha$ large enough that every step contracts a curve C with $P\cdot C=0$. Ranging $\lambda\to 0$ gives an MMP with scaling whose steps are all P-trivial. The core discovery is that, under the weak nonvanishing conditions $P\equiv D\ge 0$ or $M\equiv D\ge 0$, such a P-trivial MMP degenerates to an ordinary MMP on $K_X+B+M+\alpha P$ (Theorem 4.22), and any such degenerating MMP terminates with a minimal model (Lemma 4.30). This establishes Conjecture 1.1—that a birationally nef Nakayama-Zariski positive part forces a minimal model—in the g-klt case for all dimensions and the g-lc case in dimension 3; in dimension 4 the same conclusion holds when the data is log abundant. The results hold when the data $M'$ is merely a nef $\mathbb{R}$-divisor, the non-NQC regime where numerical nonvanishing can fail, as Example 1.2 shows.

Load-bearing premise

The P-trivial MMP is built on running an MMP on $K_X+(B+\lambda A+M)+\alpha P$ that terminates by [BCHM10], a theorem for usual klt pairs; the paper does not reduce the non-NQC generalised pair, where $M'$ is only a nef $\mathbb{R}$-divisor, to that setting, so if that auxiliary MMP cannot be run or terminated for non-NQC data, the construction and the main theorems collapse.

Editorial extensions

If this is right

  • Theorem 1.4: projective generalised klt pairs, in any dimension, whose log canonical divisor birationally has a Nakayama-Zariski decomposition with nef positive part and which satisfy $K_X+B+M\equiv D\ge 0$ or $M\equiv D\ge 0$, admit a minimal model.
  • Theorem 1.5: the same holds for generalised lc pairs in dimension 3, where the special termination theorem and degeneration criterion are available unconditionally.
  • Theorem 1.6: in dimension 4, minimal models exist for generalised lc pairs whose data $M'$ is log abundant, assuming the positive part is nef birationally.
  • Conditional higher-dimensional statements: if Conjecture 4.23 holds in dimension $d-1$, then g-lc pairs of dimension $d$ with log effective data or positive part have minimal models (Theorem 4.31); if terminations for usual dlt MMP hold up to dimension $d-1$, log abundance of data gives minimal models (Theorem 4.33).
  • Theorem 4.37: under the same log effective hypotheses, every projective g-lc pair has a weak minimal model, and if $B-(K_X+B+M)$ avoids the g-lc centres, a minimal model.

Reading between the lines

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

  • The P-trivial MMP gives a substitute for rational structure in non-NQC problems: wherever NQC was previously needed just to control intersection numbers, a boundedness result for $P\cdot C$ might suffice, which could make the technique applicable to abundance or contraction theorems for non-NQC g-pairs, not just minimal-model existence.
  • The paper's dimension-3 result suggests that Conjecture 4.23—that any degenerating P-trivial MMP terminates—may be approachable by induction using special termination (Theorem 4.19), since the induction step already works under the 'log effective' hypothesis; verifying the conjecture in dimension 3 would give a full dimension-4 statement in the style of Theorem 1.5 without the log-abundance restri
  • One could test whether the weak nonvanishing hypotheses can be relaxed to 'the set $\{P\cdot C\}$ is bounded away from zero along extremal curves', which is what the P-trivial construction really uses; if so, the theorems would extend to pairs where P is merely nef with uniformly positive intersection against $K$-negative extremal rays.
  • The elliptic-curve example (Example 1.2) shows that without an effectivity hypothesis numerical nonvanishing can fail, but the paper's results indicate that the obstruction is not to minimal models themselves but to the rational structure of the nef cone; a natural next step is to check whether the same P-trivial method yields Mori fibre spaces when $K_X+B+M$ is not pseudo-effective.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The author develops a theory of P-trivial MMP: for a Q-factorial generalised dlt pair and a nef divisor P, one runs an MMP on $K_X+(B+\lambda A+M)+\alpha P$ with scaling of a b-ample divisor A, choosing $\alpha$ large enough that every step contracts a curve C with $P\cdot C=0$. Ranging $\lambda\to 0$ gives an MMP with scaling whose steps are all P-trivial. The core discovery is that, under the we

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a theory of P-trivial MMP for generalised pairs (X,B+M) whose data M' is only assumed to be a nef R-divisor, and uses it to prove existence of minimal models under weak nonvanishing, log effectiveness, or log abundance assumptions. The main results are Theorem 1.5 (g-lc threefolds), Theorem 1.6 (g-lc fourfolds with log abundant data), Theorem 1.11 (conditional on Conjecture 4.23), Theorem 1.12 (conditional on usual dlt termination), and Theorem 1.13 on weak minimal models. The central new tool is the P-trivial MMP of Definition 4.5, which is defined by running an MMP on K_X+(B+λA+M)+αP and invoking [BCHM10] for existence and termination.

Significance. If the technical gap identified below can be repaired, this would be a significant contribution: it would extend the Birkar–Hu results [BH14] to generalised pairs with non-NQC nef data, introduce a useful P-trivial MMP framework with a special termination statement (Theorem 4.19) and a degeneration criterion (Theorem 4.22), and give unconditional minimal-model results in dimensions 3 and 4 under natural positivity hypotheses. The paper is honest about its conditional statements and includes Example 1.2 illustrating the non-NQC phenomenon. However, in its current form the central construction is not self-contained.

major comments (2)
  1. [Definition 4.5 (and Definition 1.7)] This definition asserts that, for every 0<λ≪1, there is α≫0 such that an MMP on K_X+(B+λA+M)+αP can be run and terminates with a good minimal model or Mori fibre space 'by [BCHM10]'. This is not justified. [BCHM10] applies to usual klt pairs (X,Δ) with an effective R-boundary Δ; here M is only a nef R-divisor, not assumed NQC or effective, and P is only nef. The support condition Supp{P}∪Supp{M}⊆Supp{B+A} controls supports but does not imply that K_X+(B+λA+M)+αP is numerically equivalent to K_X+Δ for an effective boundary Δ with coefficients <1. Example 1.2 shows that non-NQC nef divisors in this setting can have irrational coefficients and negative coefficients on components. The existence and termination of this MMP is precisely a non-NQC generalised-pair MMP statement that is not proved in the paper and is not covered by [BCHM10]. Since the P-trivial MMP is the engine behind Theorem 4.19, Theorem 4.22, Lemma 4.30, and hence Theorems 4.31, 4.33, 4.37 and Corollaries 4.32, 4.34, the main theorems are unsupported unless this input is supplied. The paper's own Question 4.6, which asks when a P-trivial MMP terminates, underscores that termination is not established by the surrounding results.
  2. [Lemma 4.1 and Corollaries 3.20–3.21] Lemma 4.1 defines λ(α) and asserts that if one runs an MMP on K_X+B+M+lP with scaling of C, then each step is P-trivial; but it does not prove that such an MMP exists, since it assumes the availability of contractions and flips for a non-NQC g-pair. Corollary 3.20 likewise proves only P-triviality of steps, not their existence. Nevertheless, the proof of Corollary 3.21 says 'By Corollary 3.20 we can run an MMP on K_X+B+M+αP'; the existence of that MMP is not supplied. The same gap appears in the proof of Theorem 3.23. This is a second instance of the same load-bearing problem: the paper repeatedly assumes that an MMP with scaling can be run for non-NQC generalised pairs, whereas this is part of what needs to be proved.
minor comments (4)
  1. [Corollary 3.21] The statement says 'Then, (X,B+M) is a minimal model'; it should say 'has a minimal model'.
  2. [Theorem 3.23 and Definition 4.20] There are typos: 'termminates' in Theorem 3.23 and 'degenrates' in Definition 4.20.
  3. [Lemmas 3.9–3.11] These lemmas rely on the unpublished manuscript [Hu21]; since they are used to establish the equivalence of g-dlt definitions and to construct small Q-factorialisations, the relevant statements should be made available or reproduced in the paper.
  4. [Definition 4.11] In the definition of the local Cartier index, the phrase 'Pick d−2 general hyperplanes H_i passing through x' is a bit imprecise because x is the generic point of a codimension-two subvariety; specifying that the H_i are general hyperplanes through the closure of x would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from external MMP inputs (BCHM10, Bir12, Nak04), not from the conclusions they aim to prove.

full rationale

The derivation chain is not circular. The P-trivial MMP of Definition 4.5 is built by running, for each small λ, an MMP on K_X+(B+λA+M)+αP whose termination is imported from the external theorem [BCHM10]; the paper does not reduce the non-NQC generalised pair to a usual klt pair, so that is a fragile or missing justification, but it is an unsupported external input rather than a conclusion used as a premise. The main theorems then use the P-trivial MMP to contract the negative part Nσ(K_X+B+M) (Lemma 4.30), and the claimed minimal models are not assumed at the outset. Self-citations [Hu21] and [Hu17] supply auxiliary technical facts about g-dlt pairs, small Q-factorialisations, and the definition of weak minimal models; in the quoted text the key lemmas are either proved in the present paper (Lemmas 3.9, 3.10, Proposition 3.11) or supported by external references such as [KM98], so the central claim does not reduce to an unverified self-citation. There are no fitted parameters being relabelled as predictions, no uniqueness theorem imported from the authors' own prior work to force a choice, and no known result being renamed as a new structure. The paper's own flagged limitations (Definition 4.5's appeal to [BCHM10] for non-NQC data, Questions 4.6 and 4.8, Remark 4.21, and the conditional Conjecture 4.23) are honest statements of missing inputs; they are correctness risks, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof is a pure mathematics proof built on standard MMP theorems. The central claim depends on the external [BCHM10] machinery, on the author's unpublished preprint [Hu21] for g-dlt structure, and on explicitly assumed termination conjectures in higher dimensions. No empirical parameters or new physical-style entities are introduced.

assumptions (6)
  • domain assumption Existence and termination of MMP with scaling for the relevant pairs as invoked in Definition 4.5
    The P-trivial MMP runs an MMP on K_X+(B+lambda A+M)+alpha P and terminates by [BCHM10]; BCHM10 concerns usual klt pairs, and no reduction for non-NQC generalised data is provided.
  • standard math BCHM10 minimal model theorem for usual klt pairs
    Imported as background from [BCHM10], used throughout the paper for existence of MMPs and minimal models in the usual pair setting.
  • domain assumption Lemmas 3.9 to 3.12 from the unpublished preprint [Hu21]
    Structural facts about g-dlt pairs, small Q-factorialisations and normal lc centres are quoted from the author's unpublished work without proof in this paper.
  • domain assumption Conjecture 4.23 in dimension d-1
    Explicitly assumed in Theorem 4.31 for the higher-dimensional result; the 3-fold case is unconditional because the conjecture is proved in dimension 2 as Lemma 4.24.
  • domain assumption Termination of MMP for dlt usual pairs in dimension at most d-1
    Assumed in Theorem 4.33; for dimension 4 the assumption is known, giving Corollary 4.34.
  • standard math Nakayama-Zariski decomposition exists and satisfies standard properties
    Imported from [Nak04] and used to define the positive and negative parts of pseudo-effective divisors.

how reviews work

0 comments
Cite this review

Pith. "Pith review of $P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs." pith.science (2026). https://pith.science/paper/D5UWX77O

@misc{pith2026250107551,
  author       = {Pith},
  title        = {Pith review of: $P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D5UWX77O}},
  note         = {Machine review of arXiv:2501.07551}
}
abstract

We develop a theory of $P$-trivial MMP whose each step is $P$-trivial for a given nef divisor $P$. As an application, we prove that, given a projective generalised klt pair $(X,B+M)$ with data $M'$ being just a nef $\mathbb{R}$-divisor, if $K_X+B+M$ birationally has a Nakayama-Zariski decomposition with nef positive part, and either if $M'$ or the positive part is log numerically effective, then it has a minimal model. Furthermore, we prove this for generalised lc pairs in dimension $3$. This is a generalisation of the main theorem of [Birkar-Hu14]. We also prove some related results.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 23 canonical work pages

  1. [1]

    Abramovich, A

    D. Abramovich, A. J. de Jong, Smoothness, semistability, and toroidal geometry. J. Algebraic Geom. 6 (1997) (4), 789--801

    work page 1997

  2. [2]

    Abramovich, F

    D. Abramovich, F. Oort, Alterations and Resolution of Singularities, Resolution of Singularities. Progress in Mathematics, 181 (2000), pp 39--108

    work page 2000

  3. [3]

    Birkar, Singularities on Fano fibrations and beyond, arXiv:2305.18770

    C. Birkar, Singularities on Fano fibrations and beyond, arXiv:2305.18770

    arXiv

  4. [4]

    Birkar, Generalised pairs in birational geometry, EMS Surv

    C. Birkar, Generalised pairs in birational geometry, EMS Surv. Math. Sci. 8 (2021), 5--24

    work page 2021

  5. [5]

    Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann

    C. Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. 193 (2021), no. 2, 347--405

    work page 2021

  6. [6]

    Birkar, Anti–pluricanonical systems on Fano varieties

    C. Birkar, Anti–pluricanonical systems on Fano varieties. Ann. of Math. (2) 190 (2) 345 - 463, September 2019. https://doi.org/10.4007/annals.2019.190.2.1

    doi:10.4007/annals.2019.190.2.1 2019

  7. [7]

    Birkar, Existence of log canonical flips and a special LMMP, Publ

    C. Birkar, Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes \'Etudes Sci. 115 (2012), no. 1, 325--368

    2012

  8. [8]

    Birkar, P

    C. Birkar, P. Cascini, C. D. Hacon, J. M c Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405--468

    2010

Show all 43 references
  1. [9]

    Birkar, Z

    C. Birkar, Z. Hu, Polarized pairs, log minimal models, and Zariski decompositions, Nagoya Math. J. 215 (2014), 203--224

    2014

  2. [10]

    Birkar, D

    C. Birkar, D. Q. Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Publ. Math. Inst. Hautes \'Etudes Sci. 123 (2016), no. 1, 283--331

    2016

  3. [11]

    Minimal model program for algebraically integrable foliations and generalized pairs, preprint arXiv:2309.15823

    arXiv

  4. [12]

    B. Chen, J. Liu, and L. Xie, Vanishing theorems for generalized pairs, arXiv:2305.12337

    arXiv

  5. [13]

    Choi, S Jang, DW Lee, Plc pairs with good asymptotic base loci, preprint arXiv:2411.04628

    S. Choi, S Jang, DW Lee, Plc pairs with good asymptotic base loci, preprint arXiv:2411.04628

    arXiv

  6. [14]

    Choi and J

    S. Choi and J. Park, Potentially non-klt locus and its applications, Math. Ann. 366, (2016), no. 1, 141–166

    2016

  7. [15]

    Das and C

    O. Das and C. D. Hacon, On the Minimal Model Program for K\"ahler 3-Folds, arXiv:2306.11708

    arXiv

  8. [16]

    Das and C

    O. Das and C. D. Hacon, Transcendental Minimal Model Program for Projective Varieties, arXiv:2412.07650

    arXiv

  9. [17]

    O. Das, C. D. Hacon, and J. Yáñez, MMP for Generalized pairs on Kähler 3-Folds, arXiv:2305.00524

    arXiv

  10. [18]

    Demailly, C

    J.-P. Demailly, C. D. Hacon, M. P a un, Extension theorems, non-vanishing and the existence of good minimal models, Acta Math. 210 (2013), no. 2, 203--259

    2013

  11. [19]

    On subadditivity of the logarithmic Kodaira dimension

    O. Fujino; Corrigendum to "On subadditivity of the logarithmic Kodaira dimension". J. Math. Soc. Japan 72 (2020), no. 4, 1181--1187

    2020

  12. [20]

    Fujino; Special termination and reduction to pl flips

    O. Fujino; Special termination and reduction to pl flips. In Flips for 3-folds and 4-folds, Oxford University Press (2007)

    2007

  13. [21]

    Jang; Anticanonical minimal models and Zariski decomposition

    S. Jang; Anticanonical minimal models and Zariski decomposition. Preprint. arXiv:2405.10533

    arXiv

  14. [22]

    C. D. Hacon, J. M c Kernan, Existence of minimal models for varieties of log general type, I\!I

  15. [23]

    C. D. Hacon, J. M c Kernan, C. Xu, Boundedness of moduli of varieties of general type,

  16. [24]

    C. D. Hacon, J. M c Kernan, C. Xu, On the birational automorphisms of varieties of general type, Ann. of Math. 177 (2013), no. 3, 1077–1111

    2013

  17. [25]

    C. D. Hacon and M. Păun, On the Canonical Bundle Formula and Adjunction for Generalized Kaehler Pairs, arXiv:2404.12007

    arXiv

  18. [26]

    J. Han, W. Liu, On Numerical Nonvanishing for Generalized Log Canonical Pairs. Doc. Math. 25 (2020), pp. 93–123. DOI 10.4171/DM/739

    2020 doi

  19. [27]

    J. Han, Z. Li, Weak Zariski decompositions and log terminal models for generalized polarized pairs. Preprint (2018)

    2018

  20. [28]

    Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No

    R. Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No. 52 . Springer-Verlag, New York-Heidelberg, 1977

    1977

  21. [29]

    Hashizume, Non-vanishing theorem for generalized log canonical pairs with a polarization, Selecta Mathematica 28 (4) (2022), 77

    K. Hashizume, Non-vanishing theorem for generalized log canonical pairs with a polarization, Selecta Mathematica 28 (4) (2022), 77

    2022

  22. [30]

    Hashizume, Z

    K. Hashizume, Z. Hu, On minimal model theory for log abundant lc pairs, J. Reine Angew. Math., 767 (2020), 109--159

    2020

  23. [31]

    Z. Hu, Existence of canonical models for Kawamata log terminal pairs, Birational Geometry, K\"ahler-Einstein Metrics and Degenerations, Springer Proceedings in Mathematics and Statistics, vol 409. Springer, Cham. https://doi.org/10.1007/978-3-031-17859-7-15

    doi

  24. [32]

    Hu, An abundance theorem for generalised pairs, preprint (2021), arXiv:2103.11813

    Z. Hu, An abundance theorem for generalised pairs, preprint (2021), arXiv:2103.11813

    2021 arXiv

  25. [33]

    Hu, Log abundance of the moduli b-divisors of lc-trivial fibrations, preprint (2020), arXiv:2003.14379

    Z. Hu, Log abundance of the moduli b-divisors of lc-trivial fibrations, preprint (2020), arXiv:2003.14379

    2020 arXiv

  26. [34]

    Hu, Log canonical pairs with boundaries containing ample divisors, preprint (2017), arXiv:1712.07219

    Z. Hu, Log canonical pairs with boundaries containing ample divisors, preprint (2017), arXiv:1712.07219

    2017 arXiv

  27. [35]

    Koll\'ar, S

    J. Koll\'ar, S. Mori, Birational geometry of algebraic varieties , With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134 . Cambridge University Press, Cambridge, (1998)

    1998

  28. [36]

    Lazarsfeld, Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series , (2004)

    R. Lazarsfeld, Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series , (2004). 10.1007/978-3-642-18808-4

    2004 doi

  29. [37]

    Liu and L

    J. Liu and L. Xie, Relative Nakayama-Zariski decomposition and minimal models of generalized pairs, Peking Math. J. (2023)

    2023

  30. [38]

    Liu and L

    J. Liu and L. Xie, Semi-ampleness of generalized pairs, Adv. Math. 427 (2023), 109126

    2023

  31. [39]

    Nakayama, Zariski-decomposition and abundance , MSJ Mem., 14 , Mathematical Society of Japan, Tokyo, (2004)

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

    2004

  32. [40]

    Shokurov, 3-fold log flips , With an appendix by Yujiro Kawamata, Russian Acad

    V.V. Shokurov, 3-fold log flips , With an appendix by Yujiro Kawamata, Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95-202

    1993

  33. [41]

    Tsakanikas, L

    N. Tsakanikas, L. Xie; Remarks on the existence of minimal models of log canonical generalized pairs , Math. Z. 307, 20 (2024). https://doi.org/10.1007/s00209-024-03489-6

    2024 doi

  34. [42]

    van Dobben de Bruyn, Infinitely small intersections with nef -Cartier divisors

    R. van Dobben de Bruyn, Infinitely small intersections with nef -Cartier divisors. https://mathoverflow.net/questions/291154/infinitely-small-intersections-with-nef-mathbb-r-cartier-divisors

  35. [43]

    Xie, Contraction theorem for generalized pairs, Algebraic Geometry and Physics 1 (2024), no

    L. Xie, Contraction theorem for generalized pairs, Algebraic Geometry and Physics 1 (2024), no. 1, 101--124

    2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.