{"id":"90ee51d0-a0f2-4447-8ce9-e5df612194ff","arxiv_id":"2502.00790","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If the extra divisor in a potential triple admits a birational Zariski decomposition, the triple can be rewritten as a generalized pair, allowing the minimal model program to run.","lead":"This paper studies 'potential triples', a broad class of singular spaces formed by a pair plus an extra pseudoeffective divisor. It shows that when the extra divisor admits a birational Zariski decomposition, these spaces inherit the minimal model program from generalized pairs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's pointwise discrepancy identification silently assumes divisorial Zariski decompositions are preserved under further blowups; without that, pNklt(X,∆,D)=gNklt and Theorem 1.1 do not follow.","rationale":"The paper's main claim, Theorem 1.1, rests entirely on the reduction in Theorem 3.1: potential triples with a birational Zariski decomposition are identified pointwise with generalized pairs. The only non-formal step in that identification is the assertion that, after passing to a higher birational model, the coefficient of any exceptional divisor in the negative part N equals its asymptotic valuation σ_E(f*D). This is a standard compatibility property of Nakayama's divisorial Zariski decomposition, but the text neither proves it nor names the exact lemma. If the property is true, the proof is complete modulo adding a reference; if it is false, the equality pNklt=gNklt fails and both conclusions of Theorem 1.1 collapse. I found no other flaw of comparable weight: the use of [CHLX, Theorem 2.2.3] is appropriate once the associated generalized pair is glc, and the extra dependence on [CJK] in Theorem 1.3 concerns a secondary application rather than the central reduction. The reader's weakest-assumption analysis identifies the same point, and the CONDITIONAL verdict remains appropriate: no fatal error is apparent, but the missing compatibility lemma should be supplied.","tokens_in":10694,"tokens_out":27679,"duration_ms":272328,"concrete_test":"Take a smooth projective threefold Y with a divisorial Zariski decomposition D=P+N, P nef, and let g:W→Y be the blowup along a smooth curve C⊂Supp(N). Compute σ_E(g*D) for the exceptional divisor E and compare it with mult_E(g*N). If these differ, Theorem 3.1's equation is false. As a surface sanity check, repeat with the blowup of a smooth surface at a point p∈Supp(N); equality is necessary for the theorem to hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.1 equates a(E;X,∆,D)=1−mult_E(∆_Y+N) for every prime divisor E over X by 'taking Y higher if necessary.' This requires that, after replacing Y by any higher birational model g:W→Y, the negative part of the divisorial Zariski decomposition of g*f*D is exactly g*N, i.e., σ_E(g*f*D)=mult_E(g*N) for every E on W. The text only defines N as the negative part on Y and never proves this pullback compatibility, nor does it cite the specific [Nak] statement that would imply it. If σ_E were larger or smaller than mult_E(g*N), the equality between potential and generalized log discrepancies fails on exceptional divisors, so pNklt(X,∆,D) need not equal gNklt(X,(∆+f*N)+f*P), and the reduction of the potential MMP to the generalized MMP in Theorems 3.3 and 1.1 breaks. The paper cites [Nak, Theorem III.5.16] only for independence of resolution, not for compatibility of Nσ with pullbacks; a lemma should be stated, proved, or explicitly referenced before Theorem 3.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10891,"tokens_out":25796,"duration_ms":260576,"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":[{"comment":"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":"Section 3, Theorem 3.1 (proof)"},{"comment":"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.","section":"Section 3, Theorem 1.3 (proof)"}],"minor_comments":[{"comment":"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.","section":"Notation, throughout"},{"comment":"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":"Throughout"},{"comment":"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":"Section 3, Proposition 3.5 (proof)"},{"comment":"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.","section":"Section 2.5, Definition 2.4"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely correct, but the proof of the crucial reduction (Theorem 3.1) omits a nontrivial pullback compatibility statement for Nσ, and the paper leans on unpublished citations ([CJK], [CJL]) for one of its main results. If the authors supply the missing lemma and either prove or publish the needed statement from [CJK], the paper would be suitable for publication. The scope and subject matter fit the journal well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nCorollary 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.\n\nOverall: 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.\n\nBest,","headline":"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.","tokens_in":11547,"tokens_out":5062,"would_cite":true,"duration_ms":48697,"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":"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.","keywords":["potential triple","generalized pair","Zariski decomposition","minimal model program","potential log discrepancy","pklt pair","plc pair","asymptotic valuation"],"falsifier":"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.","tokens_in":10409,"feed_emoji":"","tokens_out":13308,"duration_ms":117286,"temperature":0.7,"pith_summary":"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.","feed_headline":"Zariski decomposition lets potential triples run the MMP","feed_subtitle":"Potential log discrepancies equal generalized ones, so PLC loci are closed and minimal models exist","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the generalized MMP for Q-factorial glc pairs that Theorem 1.1 invokes to run the (K_X+∆+D)-MMP","marker":"[CHLX, Theorem 2.2.3]"},{"why":"establishes that asymptotic valuations σ_E(D) are well-defined on higher models, grounding the birational Zariski decomposition","marker":"[Nak, Theorem III.5.16]"},{"why":"gives existence of minimal models for NQC gklt pairs, used in the proof of Corollary 1.2","marker":"[TX, Theorem 5.18]"},{"why":"shows the potential-triple category strictly contains generalized pairs, motivating the need for the new comparison","marker":"[CJK, Remark 3.1]"},{"why":"prior closedness of pNklt in the anticanonical case that Corollary 3.2 extends","marker":"[CJ, Theorem 2.6]"},{"why":"example where the diminished base locus is not Zariski closed, showing the birational Zariski decomposition hypothesis is not automatic","marker":"[Les]"},{"why":"the anticanonical minimal-model statement that Corollary 1.2 recovers","marker":"[Jan, Theorem 1.3]"},{"why":"gives a big divisor without birational Zariski decomposition, marking the boundary of the theorem's hypothesis","marker":"[Nak, Chapter IV, §2]"}],"fun_headline_variants":["Zariski splits unleash MMP for potential triples","Potential triples get MMP via Zariski decomposition","Generalized pairs from Zariski: MMP for triples","For pklt pairs, anticanonical Zariski yields minimal models","NQC positive part: minimal models from Zariski splits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zariski splits unleash MMP for potential triples","Potential triples get MMP via Zariski decomposition","Generalized pairs from Zariski: MMP for triples","For pklt pairs, anticanonical Zariski yields minimal models","NQC positive part: minimal models from Zariski splits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000191,"raw_usage":{"total_tokens":1376,"prompt_tokens":1010,"completion_tokens":366,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":626,"completion_tokens_details":{"reasoning_tokens":282}},"tokens_in":626,"tokens_out":366,"duration_ms":4541,"temperature":1.0,"reasoning_tokens":282,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T17:44:04.853238+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}