{"id":"817dc19a-40a7-48e6-88cb-d62542f30207","arxiv_id":"2501.07551","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A P-trivial MMP is introduced and used to show that several classes of generalised klt and lc pairs with Nakayama-Zariski decompositions have minimal models.","lead":"This paper develops a version of the minimal model program in which each step preserves a chosen nef divisor, and uses it to prove that certain generalised pairs have minimal models when their log canonical divisor has a Zariski decomposition with nef positive part. The results extend earlier work by Birkar and Hu from ordinary pairs to generalised pairs with non-rational nef data, in dimension 3 and under weaker hypotheses in higher dimensions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 4.5 invokes [BCHM10] to terminate an MMP on K_X+(B+λA+M)+αP for a non-NQC generalised pair, without reducing it to a usual klt pair; if this input is absent the P-trivial MMP and all main theorems are unsupported.","rationale":"The paper is a serious and carefully structured contribution, and the P-trivial MMP is a plausible new technique. I read it in good faith: the intended central claim is that, under weak nonvanishing assumptions, one can run an MMP whose steps are P-trivial and eventually terminate with a minimal model. The load-bearing step is the existence and termination of the MMP in Definition 4.5, attributed to [BCHM10]. My concern is not that the theorems are false, but that the cited input is not available as stated: BCHM10 concerns usual klt pairs, and the manuscript does not reduce the non-NQC generalised pair K_X+(B+λA+M)+αP to that setting. The support condition in Definition 4.5 does not by itself make M or P into boundary divisors. This is precisely the same weakest point identified by the reader. Because the reader already returned a CONDITIONAL verdict with low confidence, I do not change the verdict: the paper should be accepted only if the MMP-with-scaling input can be rigorously justified, either by an explicit reduction to a klt boundary or by a cited theorem for non-NQC generalised pairs. The proposed concrete test would settle whether the missing reduction actually exists. I also note secondary reliance on the unpublished preprint [Hu21] and on assertions that Nσ can be taken to be a Q-Cartier Q-divisor, but the Definition 4.5 gap is the primary obstacle.","tokens_in":32417,"tokens_out":16492,"duration_ms":182151,"concrete_test":"Take the non-NQC surface of Example 1.2, with X=E×E and M=F_1+√2F_2+(√2−2)Δ. Try to instantiate Definition 4.5 literally: choose a boundary B and ample A such that (X,(B+A)+M) is g-dlt and Supp P⊆Supp(B+A), with P=M. Write the claimed MMP divisor K_X+λA+M+αP. Then check whether there exists a klt boundary Δ with K_X+Δ ≡ K_X+λA+M+αP, as BCHM10 would require. If no such effective boundary Δ with coefficients <1 can be exhibited, the cited termination theorem is inapplicable. If a reduction is claimed, write it out explicitly and verify the coefficients on the components of Example 1.2, especially where M has coefficient <0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 4.5 (and Definition 1.7) asserts that for a Q-factorial g-dlt pair one can run an MMP on K_X+(B+λA+M)+αP which is P-trivial and terminates with a good minimal model or Mori fibre space \"by [BCHM10]\". BCHM10 is a theorem for usual klt pairs (X,Δ). Here M is only a nef R-divisor with data M' not assumed NQC, and P is nef; neither is shown to be absorbed into a klt boundary. The support condition Supp{P}∪Supp{M}⊆Supp{B+A} only controls where these divisors are supported; it does not make M or P effective, nor does it imply K_X+B+λA+M+αP is numerically equivalent to K_X+Δ for an effective boundary Δ with coefficients <1. Example 1.2 explicitly exhibits a nef R-divisor with irrational coefficients and a negative coefficient on a component, so the non-NQC phenomenon is real in this paper's context. Without a reduction to the usual klt MMP, the existence and termination of the MMP in Definition 4.5 is exactly the kind of non-NQC generalised MMP theorem that is not known and is not proved here. The P-trivial MMP is the engine behind special termination (Theorem 4.19), degeneration (Theorem 4.22), and the final theorems (Theorem 4.31, Corollary 4.32, Theorem 4.37). If this MMP-with-scaling step is unavailable, the central claims are unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":32694,"tokens_out":8666,"duration_ms":86174,"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":[{"comment":"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.","section":"Definition 4.5 (and Definition 1.7)"},{"comment":"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.","section":"Lemma 4.1 and Corollaries 3.20–3.21"}],"minor_comments":[{"comment":"The statement says 'Then, (X,B+M) is a minimal model'; it should say 'has a minimal model'.","section":"Corollary 3.21"},{"comment":"There are typos: 'termminates' in Theorem 3.23 and 'degenrates' in Definition 4.20.","section":"Theorem 3.23 and Definition 4.20"},{"comment":"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.","section":"Lemmas 3.9–3.11"},{"comment":"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.","section":"Definition 4.11"}],"recommendation":"major_revision","confidential_remarks":"The main unresolved issue is Definition 4.5, where [BCHM10] is invoked for a non-NQC generalised pair without reducing to a usual klt pair. This is not a small gap: it affects the existence of the P-trivial MMP and therefore the unconditional claims in the abstract. I recommend asking the authors either to provide a complete proof of the MMP statement in Definition 4.5 under their hypotheses or to restate the main theorems with an explicit additional assumption. I also note the heavy reliance on the unpublished manuscripts [Hu17] and [Hu21] for foundational lemmas; the editor may wish to request that these be made publicly available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the P-trivial MMP is a genuinely new technique, and it does real work: special termination for non-NQC g-pairs, an unconditional 3-fold g-lc minimal model theorem, and conditional higher-dimensional statements. Second, the engine rests on a step I cannot verify: Definition 4.5 claims we can run an MMP on K_X+(B+λA+M)+αP and terminate by [BCHM10], but [BCHM10] is a theorem for usual klt pairs, and the paper never reduces the non-NQC generalized pair to that setting. I agree with the stress-test note: this is not a nitpick. All main theorems pass through this definition.\n\nWhat is good: the paper is well organized and the statements are honest about what is conditional. The extension beyond [BH14] (M=0) and [TX24] (NQC) is substantial. Section 3's P-triviality lemmas (Lemma 3.8, Theorem 3.18) look correct and are useful in their own right. The special termination theorem (Theorem 4.19) is clever, assuming the P-trivial MMP exists.\n\nThe soft spot is primary. As the stress-test says, the support condition Supp{P}∪Supp{M}⊆Supp{B+A} does not make M effective or bounded; Example 1.2 shows M can have irrational and negative coefficients. So K_X+B+λA+M+αP is not a usual klt pair in any obvious way. Lemma 4.1 gives an MMP with scaling for non-NQC g-pairs, but under stronger support assumptions (Supp M and Supp P contained in Supp B) and it does not prove termination. Lemma 3.8 only proves P-triviality of an MMP assumed to exist. So Definition 4.5's appeal to BCHM10 is unsupported by anything in the paper. The author also leans on unpublished preprints [Hu21] and [Hu17], and Lemma 4.30's termination claim is compressed. These are secondary; the Definition 4.5 issue is what matters.\n\nMy verdict: send it to a serious referee. The idea is strong and the results, if correct, are important. But the referee should be instructed to focus on Definition 4.5: is there a known way to run this MMP, or is the author silently relying on a theorem that needs proof? I suspect the gap is real, but I could be wrong if there is a standard reduction the author omitted. I would not cite it yet; I would bring it to a reading group if you want a lively discussion about non-NQC g-pairs.","headline":"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.","tokens_in":33297,"tokens_out":5845,"would_cite":false,"duration_ms":56132,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30","14J17"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["generalised pairs","minimal model program","P-trivial MMP","Nakayama-Zariski decomposition","non-NQC nef divisors","special termination","log abundance","birational geometry"],"falsifier":"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.","tokens_in":32151,"feed_emoji":"📐","tokens_out":11296,"duration_ms":95049,"temperature":0.7,"pith_summary":"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.","feed_headline":"Minimal models exist for generalised 3-folds with nef-only data","feed_subtitle":"A P-trivial MMP, whose steps keep P nef, covers the case where numerical nonvanishing fails.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The main theorem this paper generalises; introduced the MMP on $K+B+\\alpha P$ approach for polarized pairs with $M=0$.","marker":"[BH14]"},{"why":"Provides the termination of the auxiliary MMP on $K_X+(B+\\lambda A+M)+\\alpha P$ used to construct P-trivial MMP steps.","marker":"[BCHM10]"},{"why":"Defines the Nakayama-Zariski decomposition ($P_\\sigma$, $N_\\sigma$) that gives the nef positive part throughout.","marker":"[Nak04]"},{"why":"Supplies Example 1.2 showing numerical nonvanishing fails for non-NQC g-pairs, motivating the weak nonvanishing hypotheses.","marker":"[HL20]"},{"why":"Prior minimal-model result requiring both M and P to be NQC; the paper weakens this to one of them.","marker":"[TX24]"},{"why":"Gives Lemmas 3.1–3.2 on contractions of very exceptional divisors, used in the degeneration and termination arguments.","marker":"[HL18]"},{"why":"Provides Lemma 3.13 and Corollary 3.15 connecting termination with scaling to existence of minimal models.","marker":"[Bir12]"},{"why":"Supplies the g-dlt structure lemmas (existence of small Q-factorialisations, normality of g-lc centres) used in special termination.","marker":"[Hu21]"}],"fun_headline_variants":["P-trivial MMP yields minimal models for generalised pairs","Nef-only data gives minimal models in dimension 3","P-trivial steps tame MMP when nonvanishing fails","Minimal models for generalised pairs from nef-only data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["P-trivial MMP yields minimal models for generalised pairs","Nef-only data gives minimal models in dimension 3","P-trivial steps tame MMP when nonvanishing fails","Minimal models for generalised pairs from nef-only data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000688,"raw_usage":{"total_tokens":3128,"prompt_tokens":966,"completion_tokens":2162,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":2094}},"tokens_in":582,"tokens_out":2162,"duration_ms":15597,"temperature":1.0,"reasoning_tokens":2094,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:39:22.749670+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}