{"id":"05a4c275-dbba-487f-9f34-74eeec778f8c","arxiv_id":"1908.06945","paper_version":4,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Assuming the Minimal Model Program in dimension n-1, every uniruled but not rationally connected projective log canonical pair of dimension n with pseudoeffective log canonical divisor has a good model.","lead":"This paper proves that every uniruled algebraic variety that is not rationally connected has a 'good model', as long as the Minimal Model Program is known in one dimension lower. It is a major step toward the Abundance conjecture, one of the central open problems in higher-dimensional geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 2 of Theorem A needs a good model for the rationally connected fibre F of the MRC fibration, but the stated hypothesis covers only non-uniruled klt pairs; the proof relies on an unproved upgrade to all lc pairs in dimension ≤ n−1 via [LM19].","rationale":"The reader correctly identified the lower-dimensional MMP hypothesis as the main caveat. My review sharpens this: the proof of Theorem A applies that hypothesis to the general fibre F of the MRC fibration, which is rationally connected rather than non-uniruled. The text attempts to upgrade the hypothesis globally at the start of Section 3, citing [LM19, Theorem 1.3 and Lemmas 2.3 and 2.4]. That upgrade is load-bearing: without it, Step 2 cannot invoke [HX13, Theorem 2.12] to obtain a relative good model over Y, and the proof collapses. The paper does not supply the derivation, so the reader cannot verify whether the upgrade is a theorem or an unsupported strengthening. This is not a disagreement with the abundance conjecture; it is an internal reliance on a stronger input than the one declared in Theorem A. The conditional verdict reflects that the claim is plausible and the proof is coherent modulo this point, but acceptance should wait until the author justifies the 'we may assume' sentence or states Theorem A with the stronger hypothesis.","tokens_in":12787,"tokens_out":21970,"duration_ms":234783,"concrete_test":"Check the exact statements of [LM19, Theorem 1.3 and Lemmas 2.3 and 2.4] and independently re-derive the sentence at the start of Section 3: 'we may assume the existence of good models for log canonical pairs in dimensions at most n−1'. The test passes only if these references prove that the stated hypothesis (good models for non-uniruled klt pairs with rational boundaries in dimension n−1) is enough to obtain good models for all log canonical pairs of dimension ≤ n−1, including rationally connected general fibres of MRC fibrations. If they prove only nonvanishing or only good models for non-uniruled pairs, then Theorem A, Step 2, is not justified as written.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem A, Step 2, the author needs a relative good model over Y in order to apply [HX13, Theorem 2.12], and asserts that the general fibre (F, Δ|F) has a good model 'by the assumption in lower dimensions'. But F is a general fibre of an MRC fibration, hence rationally connected and in particular uniruled. The theorem's stated hypothesis is only the existence of good models for non-uniruled klt pairs with rational boundaries in dimension n−1. The proof tries to bridge this gap with the first sentence of Section 3: 'By [LM19, Theorem 1.3 and Lemmas 2.3 and 2.4] we may assume the existence of good models for log canonical pairs in dimensions at most n−1.' This silently strengthens the hypothesis to all log canonical pairs of dimension < n, including rationally connected/uniruled pairs and real boundaries. The whole argument for the fibre F depends on this strengthening: without it, the stated lower-dimensional assumption does not cover F. The manuscript does not reproduce the derivation from [LM19], so it is not verifiable from the text alone. If [LM19] proves only nonvanishing for uniruled pairs, or only good models for non-uniruled pairs, then Theorem A, Step 2, has a genuine gap: the relative MMP step has no justified input and the central claim is unsupported for exactly the rationally connected fibres that the MRC fibration produces.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that a projective log canonical pair (X,Δ) of dimension n whose underlying variety X is uniruled but not rationally connected has a good model whenever K_X+Δ is pseudoeffective, assuming the existence of good models for non-uniruled klt pairs with rational boundaries in dimension n−1. The proof uses the MRC fibration, the nonvanishing theorem for uniruled log canonical pairs, subadditivity of Kodaira dimension, and a reduction to lower-dimensional cases via recent work of the author and others. The paper also proves a generalisation (Theorem C) for pairs admitting a dominant rational map to a non-uniruled base, and a conditional result (Theorem D) for rationally connected pairs subject to a newly formulated Nonexistence Conjecture.","tokens_in":13145,"tokens_out":11701,"duration_ms":101694,"significance":"The main theorems provide a substantial advance on the abundance conjecture for a large class of uniruled pairs, going beyond previous results that required fibrations over abelian varieties or other special structures. The paper is carefully written and explicitly separates the new arguments from the quoted machinery. The reduction of the rationally connected case to a very specific Nonexistence Conjecture is a useful conceptual contribution, and the paper also gives a clean proof for rationally connected pairs with κ(X,Δ)>0 or κ(X,−K_X)>0. The proofs are coherent and the conditional hypotheses are stated precisely.","major_comments":[],"minor_comments":[{"comment":"The assertion that by [LM19, Theorem 1.3 and Lemmas 2.3 and 2.4] one may assume the existence of good models for all log canonical pairs in dimensions at most n−1 is essential for Step 2, where it is applied to the rationally connected general fibre F of the MRC fibration. Since the stated hypothesis of Theorem A covers only non-uniruled klt pairs with rational boundaries, it would be helpful to state the precise result from [LM19] that justifies this strengthening, or to give a short indication of the derivation.","section":"Section 3, proof of Theorem A, opening sentence"},{"comment":"The sentence 'Since K_F+Δ|F is pseudoeffective, we have κ(F, K_F+Δ|F) ≥ 0 by the assumption in lower dimensions' is slightly imprecise, because the lower-dimensional assumption is about existence of good models rather than about nonvanishing. Clarifying that this follows from the upgraded assumption introduced at the start of the proof would avoid potential confusion.","section":"Section 3, Step 2 of Theorem A"},{"comment":"The parameter δ in the displayed relation K_X+δΔ ∼Q 0 is not explicitly defined. For clarity, one should note that δ = 1 − d_1/δ_1, which is indeed between 0 and 1 in the situation at hand.","section":"Section 4, Step 4 of Theorem D"},{"comment":"Some equations and references are cited without precise theorem numbers in a few places (e.g., the use of [KP17, Theorem 9.9] and [Nak04, Corollary V.1.12]); adding the exact statements would improve the reader's ability to verify the arguments.","section":"Throughout"}],"recommendation":"accept","confidential_remarks":"The correctness of the main results depends on several very recent works by the author and coauthors ([LM19], [LT19], [Has19], [HH19]), which were preprints at the time of writing. The editor may wish to confirm that these references have been or are being refereed and are in stable form, so that the conditional results are supported by the literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nHere's my take on Lazi'c's Abundance paper. The headline: it's a serious, short paper that proves Abundance for uniruled but not rationally connected lc pairs, conditional on the MMP in one dimension lower. The main new ingredient is the MRC fibration step: either the fibre has non-maximal Kodaira dimension and you run a relative MMP, or it has maximal Kodaira dimension and subadditivity contradicts kappa = 0. That's genuinely new; [DL15] only got (X, tau Delta).\n\nThe paper is also honest about what it does not do. Theorem D for the rationally connected case is conditional on a nonexistence conjecture, and the author says so. The real-coefficient case is handled by decomposing the boundary, a standard but delicate step, and it's done correctly. Corollary B gives an unconditional dimension-4 result, which is a nice concrete payoff.\n\nNow the soft spots. The stress-test note has a point: Step 2 of Theorem A needs a good model for the general fibre F of the MRC fibration, which is rationally connected. The stated hypothesis only gives good models for non-uniruled klt pairs with rational boundaries in dimension n-1. The proof bridges this with one sentence: 'By [LM19, Theorem 1.3 and Lemmas 2.3 and 2.4] we may assume the existence of good models for log canonical pairs in dimensions at most n-1.' That upgrade is doing real work. I think it's plausible - uniruled klt pairs are easier, and [LM19] is precisely about nonvanishing and good models there - but the paper doesn't reproduce it, and [LM19] is the author's own. A referee should check that citation line carefully. If the upgrade fails, Step 2 collapses. Everything else in the proof is built from cited results with clean logic.\n\nThere's also the general point that much of the proof sits on recent deep results (KP17, LT19, LM19). That's normal in this field, but it means the paper's correctness is only as solid as that network. I don't see circularity or overclaim.\n\nMy bottom line: this deserves serious peer review. The central theorem is a real advance, the proof is a nice idea, and the one fragile spot is checkable by a knowledgeable referee. I'd send it on, and ask the referee to verify the [LM19] reduction. I would cite it myself.","headline":"A genuinely new conditional result for uniruled non-RC pairs; the one load-bearing point is a citation to the author's [LM19] that deserves careful checking.","tokens_in":751,"tokens_out":805,"would_cite":true,"duration_ms":61335,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a projective log canonical pair whose underlying variety is uniruled but not rationally connected has a good model, conditional on the Minimal Model Program in one dimension lower.","keywords":["Abundance conjecture","Minimal Model Program","good models","uniruled varieties","rationally connected varieties","log canonical pairs","semiampleness","MRC fibration"],"falsifier":"Construct a projective log canonical fourfold $X$ that is uniruled but not rationally connected with $K_X+\\Delta$ nef but not semiample; since the Minimal Model Program is known in dimension 4, such a pair would refute the unconditional Corollary B.","tokens_in":12558,"feed_emoji":"","tokens_out":8104,"duration_ms":72934,"temperature":0.7,"pith_summary":"The paper targets the Abundance conjecture, the claim that a projective log canonical pair $(X,\\Delta)$ whose canonical class $K_X+\\Delta$ is pseudoeffective has a good model—a birational model in which $K_X+\\Delta$ becomes semiample. It proves this for all pairs whose underlying variety $X$ is uniruled (covered by rational curves) but not rationally connected (points cannot be joined by rational curves), assuming the Minimal Model Program holds in dimension $\\dim X-1$. In dimension 4, where that program is known, the conclusion is unconditional. The same method also covers pairs mapping onto non-uniruled varieties or abelian varieties, and for rationally connected $X$ it reduces the remaining problem to a sharply formulated Nonexistence Conjecture.","feed_headline":"Good models exist for uniruled non-rationally connected pairs","feed_subtitle":"If the canonical class is nef, it is semiample; in dimension 4 the result is unconditional.","key_machinery":"The maximal rationally connected (MRC) fibration of $X$: a dominant rational map whose very general fibres are rationally connected and whose base is not uniruled. The proof resolves indeterminacies to make it a fibration, then splits into two cases. If a very general fibre has maximal Kodaira dimension for $K_F+\\Delta|_F$, subadditivity of Kodaira dimension forces $\\kappa(X,K_X+\\Delta)>0$, and known abundance results for positive Kodaira dimension apply. If the fibre Kodaira dimension is smaller, a relative $(K_X+\\Delta)$-MMP over the base terminates in a relative good model, and the canonical bundle formula transfers abundance to the base. For real coefficients, the argument passes through a decomposition of the divisor into finitely many rational pieces, and the non-klt part is handled by perturbing the boundary.","core_discovery":"Theorem A is the central claim: assuming good models exist for non-uniruled klt pairs with rational boundaries in dimension $n-1$, every projective log canonical pair $(X,\\Delta)$ of dimension $n$ with $X$ uniruled but not rationally connected and $K_X+\\Delta$ pseudoeffective has a good model. In particular, if $K_X+\\Delta$ is nef, then it is semiample. The stronger Theorem C replaces the MRC fibration by any dominant rational map to a normal projective variety $Y$ with $0<\\dim Y<\\dim X$ and $Y$ not uniruled, and Corollary 3.2 applies when the target is an abelian variety. For rationally connected $X$, Theorem D shows the same conclusion follows from an explicit Nonexistence Conjecture for special klt pairs of Calabi-Yau type, and Theorem 4.1 makes it unconditional when $\\kappa(X,\\Delta)>0$ or $\\kappa(X,-K_X)>0$.","pith_inferences":["The reduction to the base of the MRC fibration suggests that abundance for uniruled varieties may be governed by the Kodaira dimension of a non-uniruled base, so one testable extension is to replace rational connectedness by other fibration properties that force the base to have pseudoeffective canonical class.","The proof of Theorem D converts abundance for rationally connected pairs into a statement about nef divisors on klt pairs of Calabi-Yau type; proving the Nonexistence Conjecture would remove the last conditional assumption in the rationally connected case.","Because all ingredients are known in dimension 3, the paper's dimension-4 result is unconditional; a concrete check would be to run the same argument on explicit 4-fold examples to see whether nef non-semiample canonical divisors are ruled out."],"forward_implications":["In dimension 4, every projective log canonical pair with $K_X+\\Delta$ pseudoeffective and $X$ uniruled but not rationally connected has a good model; if $K_X+\\Delta$ is nef, it is semiample.","A projective log canonical pair of dimension $n$ with a dominant rational map onto a non-uniruled variety of intermediate dimension has a good model, under the same lower-dimensional MMP assumption.","A projective log canonical pair with a nontrivial rational map to an abelian variety has a good model, under the same assumption.","For rationally connected $X$, the existence of a good model follows from the paper's Nonexistence Conjecture; if either $\\kappa(X,\\Delta)>0$ or $\\kappa(X,-K_X)>0$, the good model exists without that conjecture."],"supporting_citations":[{"why":"provides the nonvanishing for uniruled pairs and the blanket reduction to lower-dimensional good models that frames Theorems A, C, and D.","marker":"[LM19]"},{"why":"supplies the existence of minimal models for log canonical pairs with pseudoeffective canonical divisor, used to reduce to the nef case.","marker":"[LT19]"},{"why":"gives the partial-abundance result and the reduction of good models for uniruled pairs to non-uniruled pairs, used in the boundary perturbation steps.","marker":"[DL15]"},{"why":"supplies subadditivity of log Kodaira dimension for fibrations, used when the MRC fibre has maximal Kodaira dimension.","marker":"[KP17]"},{"why":"provides the log Iitaka conjecture for abundant log canonical fibrations, used in the proof of Theorem C.","marker":"[Has19]"},{"why":"establishes the pseudoeffectiveness criterion for the canonical divisor that makes uniruledness detectable and shows the MRC base has pseudoeffective canonical class.","marker":"[BDPP13]"},{"why":"shows families of rationally connected varieties are themselves dominated by rational curves, forcing the MRC base not to be uniruled and its canonical class pseudoeffective.","marker":"[GHS03]"},{"why":"supplies the existence of log canonical closures and the relative good model criterion used to run the relative MMP over the MRC base.","marker":"[HX13]"},{"why":"gives good models for varieties fibered by good minimal models, yielding abundance once the Kodaira dimension is positive.","marker":"[Lai11]"},{"why":"provides the Nakayama sigma-function and numerical dimension machinery used in the rationally connected case.","marker":"[Nak04]"}],"fun_headline_variants":["Abundance for uniruled pairs lacking rational connectivity","Nef canonical semiample for uniruled non-rationally connected pairs","Uniruled pairs: good models exist if not rationally connected","MMP abundance proven for uniruled pairs without rational connectivity","Dim 4: abundance for uniruled non-rationally connected pairs unconditional"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the assumption that every non-uniruled klt pair with rational boundary in dimension $n-1$ has a good model, and for the rationally connected case also on an unproved Nonexistence Conjecture for special klt pairs of Calabi-Yau type.","fun_headline_variants_meta":{"raw":{"variants":["Abundance for uniruled pairs lacking rational connectivity","Nef canonical semiample for uniruled non-rationally connected pairs","Uniruled pairs: good models exist if not rationally connected","MMP abundance proven for uniruled pairs without rational connectivity","Dim 4: abundance for uniruled non-rationally connected pairs unconditional"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000833,"raw_usage":{"total_tokens":3605,"prompt_tokens":884,"completion_tokens":2721,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":2634}},"tokens_in":500,"tokens_out":2721,"duration_ms":21681,"temperature":1.0,"reasoning_tokens":2634,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:30:24.137315+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a projective log canonical fourfold $X$ that is uniruled but not rationally connected with $K_X+\\Delta$ nef but not semiample; since the Minimal Model Program is known in dimension 4, such a pair would refute the unconditional Corollary B.","supporting_citations":[],"review_version":1}