{"id":"b1ff86e0-0e6e-4279-8ebf-0ffa60feae51","arxiv_id":"2411.16146","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every divisorial contraction to a curve between terminal threefolds is a weighted blow-up, and all such contractions are classified for smooth curves.","lead":"Algebraic geometers classify the elementary \"divisorial contractions\" used in the minimal model program. This paper proves that every such contraction to a curve on a terminal threefold is a weighted blow-up, and lists all possibilities when the curve is smooth.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's non-Gorenstein case is proved by Proposition 3.19, which requires Y→X to be a contraction between Q-factorial terminal threefolds; the theorem omits Q-factoriality and no argument shows it is automatic.","rationale":"The reader identified the same load-bearing concern: the non-Gorenstein branch of Theorem 1.1 passes through Proposition 3.19, which adds a Q-factoriality hypothesis that is absent from the theorem statement. I agree that this is the most serious gap because it affects the central claim of the paper, not just the classification table. The ordinary double point shows that being terminal does not imply Q-factoriality, and although that example is Gorenstein, it demonstrates that the missing hypothesis cannot be dismissed as automatic from terminality alone. The MMP step in Lemma 3.18 is explicitly formulated for Q-factorial threefolds, so the bridge from the Gorenstein theorem to the non-Gorenstein case is not justified as written. Other weaknesses, such as the irreducible-preimage assumption in Proposition 3.19 and the unexpanded 'Computation shows' statements in Section 6, are real but secondary: the irreducible-preimage condition is absent from Theorem 1.1 as well, and the computations concern the classification rather than the existence part of the central theorem. Because the gap is repairable by adding a hypothesis or by proving that the missing hypothesis is automatic, the existing CONDITIONAL verdict is appropriate, and I do not change it.","tokens_in":2,"tokens_out":15758,"duration_ms":283042,"concrete_test":"For the seven non-Gorenstein terminal forms listed in Theorem 2.2, compute the divisor class group of the completed local ring (equivalently, the invariant class group of the canonical cover under the cyclic action). If any of these groups is not torsion, terminality does not imply local Q-factoriality, and Theorem 1.1 must either add a Q-factoriality hypothesis or prove that divisorial contractions to smooth curves avoid such singularities. If all local class groups are torsion, the residual test is whether Lemma 3.18's MMP step can be rerun for globally non-Q-factorial terminal threefolds; if it cannot, the theorem should be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract and Theorem 1.1 assert that every divisorial contraction to a smooth curve near a terminal threefold singularity of Cartier index r≥1 is a weighted blowup under a suitable embedding. The proof for r>1 is a one-line reference: 'By Theorem 3.6 and Proposition 3.19.' Proposition 3.19 is stated for 'Y→X be a divisorial contraction to a curve C between Q-factorial terminal threefolds,' and it further assumes that the preimage of C in the canonical cover is irreducible. Neither hypothesis appears in Theorem 1.1, and no argument is supplied that contractions to smooth curves automatically satisfy them. This is not a purely cosmetic mismatch: terminal threefold singularities are not automatically Q-factorial (the ordinary double point has infinite local class group), and while that example is Gorenstein, the paper itself treats analytic Q-factoriality as a nontrivial condition when discussing the general elephant conjecture. Lemma 3.18, which constructs the cover contraction ˜Y→˜X whose valuation is then tilted, relies on the MMP extraction [Ke92, Thm 17.10] and on the tie-breaking argument; these steps are formulated for Q-factorial terminal threefolds. Since the non-Gorenstein branch of Theorem 1.1 is exactly the case where the canonical cover is needed, the current proof does not establish the theorem as stated unless Q-factoriality (or at least analytic Q-factoriality at P) follows from the contraction hypothesis. The same comment applies to the non-Gorenstein entries of Theorem 1.2, whose proof invokes Proposition 3.19.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a valuation-theoretic 'tilting' procedure to realize divisorial contractions to curves on terminal threefolds as weighted blowups in suitable affine or quotient-ambient embeddings. Theorem 1.1 asserts such a realization for every divisorial contraction to a smooth curve near a terminal singularity of Cartier index r, with the non-Gorenstein case handled through the canonical cover. The Gorenstein case is approached via embedded orthogonal valuations, v-bases, and a finite-generation argument (Theorem 3.6), while the non-Gorenstein case is delegated to Proposition 3.19. The second half of the paper runs an explicit algorithm on cDV equations and lists normal forms in Tables 1-4, leading to Theorems 1.2 and 1.3. The main structural ideas are original and the Gorenstein realization argument has substantial content, but the non-Gorenstein branch of Theorem 1.1 currently rests on hypotheses that are not stated in the theorem, and the classification section relies heavily on unexpanded computations.","tokens_in":41461,"tokens_out":7740,"duration_ms":76364,"significance":"If the results hold, this is a substantial contribution to the explicit birational geometry of threefolds: it would unify and extend work of Mori, Cutkosky, Tziolas, and Ducat by showing that divisorial contractions to smooth curves are weighted blowups under suitable embeddings, and it would provide a classification table. The paper's framework of orthogonal valuations, v-bases, and tilting with finite termination is a genuinely useful technical contribution, and Proposition 3.4 on finite generation of the associated graded algebra is a real input that deserves attention in its own right. The proof in the Gorenstein case is supported by a coherent argument, and no circularity is apparent: the valuation mu_E is external data and the approximation argument is grounded in finite generation. However, the non-Gorenstein part of Theorem 1.1 is not proven as stated, and Theorem 1.2's classification is not verifiable in the current text because the decisive terminality and group-action checks are mostly asserted. The paper's ambitions are appropriate for a leading journal, but the version under review overstates what has been demonstrated.","major_comments":[{"comment":"Theorem 1.1 is stated for a contraction Y -> X near a terminal singularity P in X of Cartier index r, with no Q-factoriality hypothesis and no condition on the preimage of C in the canonical cover. The proof is the one-line reference 'By Theorem 3.6 and Proposition 3.19.' Proposition 3.19, however, explicitly assumes that Y -> X is a divisorial contraction to a curve between Q-factorial terminal threefolds, and it also assumes that the preimage of C on the canonical cover is irreducible. Neither hypothesis is shown to follow from the assumptions of Theorem 1.1; in particular, terminal threefold singularities are not automatically Q-factorial, and the paper itself treats Q-factoriality as a nontrivial condition in Lemma 3.18. Therefore the non-Gorenstein case of Theorem 1.1 is not established as stated. The authors should either add the missing hypotheses to Theorem 1.1 or supply a proof that they are automatic for contractions to a smooth curve.","section":"Section 3, Proposition 3.19 and proof of Theorem 1.1"},{"comment":"Theorem 1.2 is a classification statement, but its proof consists only of the sentence 'The classifications of divisorial contractions are summarized in the following Tables.' The case analysis preceding it repeatedly invokes assertions of the form 'Computation shows...' (for example in Cases VI-3, VI-4, VIII, IX-2 through X-4) to decide whether the constructed threefolds are terminal and whether a non-trivial group action is admissible. These checks are load-bearing: they determine which table rows correspond to genuine divisorial contractions and which are excluded. Since no computations, algorithms, or reproducible scripts are provided, a reader cannot verify the completeness or correctness of the classification. Please provide the missing computations or a verifiable supplement.","section":"Section 6, proof of Theorem 1.2"},{"comment":"For Cartier index r > 1, the text says that after taking the canonical cover one 'can examine the G-compatibility of the classification of Gorenstein case' and then lists the quotient cases Q1-Q4 after 'similar computation.' This step requires checking that the defining equation is semi-invariant under the chosen cyclic action and that the quotient singularities are terminal. These checks are not carried out; moreover, the text itself illustrates the subtlety in Case VI-3, where it notes that y is invariant while x is not, so the displayed f cannot be semi-invariant. The passage from the Gorenstein table to the quotient table is therefore not demonstrated. Since Theorem 1.2's non-Gorenstein statement depends exactly on this passage, it needs a full argument rather than an appeal to 'similar computation.'","section":"Section 6, non-Gorenstein cases and Table 4"}],"minor_comments":[{"comment":"The cD/3 case is printed as a quotient by 1/2(0,1,2,2); since the case is labelled index 3, this appears to be a typo for 1/3.","section":"Section 2.1, Theorem 2.2(5)"},{"comment":"The notation 'v != e_i (resp. v != e_j^i for j = 1,2)' in Lemma 2.4 is unclear; please state explicitly which vectors are excluded.","section":"Section 2.2, Lemma 2.4"},{"comment":"The displayed equation 'f = s3 delta s3 - p3(s1,s2)' appears garbled; clarify whether the first term should be delta * s3 or s3 * delta s3.","section":"Section 4, Corollary 4.2"},{"comment":"In the inductive step the text defines both r_{k+1} and q_{k+1} as r(r_k); the second definition should presumably be q(r_k).","section":"Section 4, proof of Proposition 4.1"},{"comment":"There are several typos and formatting issues, e.g., 'Cur ves' in the title, 'term inal' in the abstract, and 'whcih' in Section 5; please proofread the final version.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The gap in the non-Gorenstein case of Theorem 1.1 is not merely cosmetic: the proof invokes Proposition 3.19 under hypotheses that are absent from the theorem statement. If Q-factoriality is automatic for divisorial contractions to a smooth curve in the analytic category, a short proof should be added; otherwise the theorem should be restated with the necessary assumptions. The classification part also needs either detailed computations or a reproducible supplement before the paper can be checked. I would be willing to look at a revised version once these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my read of the paper. The core is new: the tilting algorithm, which modifies embeddings and weights until the exceptional divisor of a divisorial contraction is realized as a weighted blowup, is a promising technique that goes beyond the special cases of Mori, Cutkosky, Tziolas, and Ducat. The Gorenstein case (Theorem 3.6) and the explicit algorithm in Section 5 are real work, and the classification tables are a substantial amount of computation. If the classification is right, it settles a central open problem in the explicit MMP.\n\nThe soft spot the referee will hit immediately is the gap between Theorem 1.1 and its proof. Theorem 1.1 is stated for a terminal singularity of Cartier index r≥1, without Q-factoriality. The non-Gorenstein case is dispatched in one line: “By Theorem 3.6 and Proposition 3.19.” But Proposition 3.19 assumes that Y→X is a divisorial contraction between Q-factorial terminal threefolds and that the preimage of C in the canonical cover is irreducible. Neither assumption is in Theorem 1.1, and the authors do not argue that either follows. Terminal threefold singularities are not automatically Q-factorial – the ordinary double point is the standard example – and while that example is Gorenstein, the paper itself treats analytic Q-factoriality as a nontrivial condition around the general elephant conjecture. Lemma 3.18, which produces the cover contraction, uses [Ke92, Thm 17.10] and tie-breaking, both formulated in the Q-factorial setting. So the non-Gorenstein part of Theorem 1.1 is not proved as stated. The same gap affects the non-Gorenstein rows of Theorem 1.2 and, through them, the elephant corollary.\n\nThe other soft spot is Section 6. It is full of “computation shows” statements with almost no expansion. For a classification paper that is a much heavier referee burden, but it is not by itself a fatal flaw.\n\nThe authors do state limitations in the text – they only classify smooth curves, and for non-Gorenstein points they only treat the case where the preimage of the curve in the canonical cover is irreducible (and in practice smooth). So the abstract is a bit stronger than what is actually proved.\n\nWho is this for? Birational geometers working on the explicit MMP, especially anyone studying divisorial extractions from threefold terminal singularities. It deserves a serious referee, not a desk reject. My recommendation: send it to review, and require the authors to either prove that the contraction is automatically Q-factorial at the point or weaken Theorem 1.1 accordingly. The tilting method is likely to be useful even if the non-Gorenstein step needs an extra hypothesis.","headline":"A genuinely new approach to classifying threefold divisorial contractions to curves, but the non-Gorenstein case of the main theorem currently rests on a Q-factoriality hypothesis that is neither stated nor shown.","tokens_in":42089,"tokens_out":3541,"would_cite":false,"duration_ms":32598,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30","14J30","14E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every divisorial contraction to a curve in a terminal threefold is a weighted blow-up.","keywords":["divisorial contractions","threefolds","weighted blow-ups","terminal singularities","minimal model program","tilting algorithm","general elephant conjecture","canonical covers"],"falsifier":"Exhibit a divisorial contraction to a smooth curve whose target is a terminal non-Gorenstein threefold that is not Q-factorial and check whether its exceptional valuation can be realized by the tilting construction; if no embedding and weights exist, Theorem 1.1 as stated is false. Alternatively, check any row of the classification tables by computing the weighted blow-up and testing whether the proper transform has only terminal singularities, since a single non-terminal row would break the classification.","tokens_in":40951,"feed_emoji":"","tokens_out":6183,"duration_ms":54310,"temperature":0.7,"pith_summary":"This paper aims to show that divisorial contractions to curves between terminal threefolds are not exotic: each one is, after choosing a suitable embedding of the target into an affine space or a cyclic quotient of one, simply the weighted blow-up of that embedding. The first main theorem states this for contractions to a smooth curve near a terminal singularity of Cartier index $r \\ge 1$, with the weighted blow-up taken in $A^N/\\mu_r$. The second theorem classifies all such weighted blow-ups when the curve is smooth, listing them in tables by the normal form of the defining equation. The authors also prove the general elephant conjecture for these contractions: a surface through the contracted curve with at worst Du Val singularities exists. The motivation is explicit understanding of the minimal model program in dimension three, where contractions to points were already classified.","feed_headline":"Every curve contraction in terminal threefolds is a weighted blow-up","feed_subtitle":"A tilting algorithm finds the embedding and weights, yielding classification tables and Du Val elephants for smooth curves.","key_machinery":"The machinery is the tilting algorithm for valuations. Starting from an embedding $X \\subset A^4$ (or its cyclic quotient), the authors assign weights by the valuations of coordinate functions with respect to the exceptional divisor $E$, and compare the induced orthogonal valuation with $\\mu_E$ restricted to $X$. Whenever the two disagree, they add a new ambient variable $y_i$ and a new weight, replacing some coordinate by a section $y_i - \\kappa_i$, a process called tilting; the valuation is unchanged on the subvariety while the filtration gets strictly closer to the target. Finite generation of the relevant graded algebra (Proposition 3.4) guarantees termination after finitely many tilts. A $v$-basis of the ideal of $X$ makes the exceptional divisor computable: its irreducible components are cut out by the lowest-weight homogeneous parts of the basis, and irreducibility of this set is equivalent to the restricted valuation being an actual valuation. For the non-Gorenstein case, the canonical cover reduces the problem to a $G$-equivariant version of the same construction.","core_discovery":"At the paper's core is the claim that the exceptional divisor valuation of any divisorial contraction to a curve can be reproduced by an orthogonal valuation coming from a weighted blow-up, after enlarging the ambient space. For a contraction $\\varphi: Y \\to X$ to a smooth curve $C$ through a terminal singularity $P$ of Cartier index $r$, the authors construct an embedding $X \\subset A^N/\\mu_r$ and weights $w$ such that $Y$ is isomorphic to the proper transform of $X$ under the weighted blow-up with weights $w$ (Theorem 1.1). When $C$ is smooth and $X$ is Gorenstein terminal, or when the preimage of $C$ in the canonical cover is irreducible in the non-Gorenstein case, the weighted blow-up is one of a finite list given in the tables, with ambient dimension at most 5 (Theorem 1.2). As a corollary they obtain a surface through $C$ with Du Val singularities, proving the general elephant conjecture under these hypotheses (Theorem 1.3). The proof is constructive: the tilting algorithm produces the embedding and weights by repeatedly changing coordinates and adding variables until the induced valuation matches the given one.","pith_inferences":["The Q-factoriality assumption in Proposition 3.19 is load-bearing for the non-Gorenstein part of Theorem 1.1 but is not stated in Theorem 1.1; if Q-factoriality is not automatic for such contractions, the theorem needs an extra hypothesis or a modified proof.","The tilting algorithm suggests a computational strategy: for a given defining equation $f$, the termination step and the resulting weights can be computed symbolically, which could turn the classification into an automated check for new examples.","The same valuation-matching technique may transfer to divisorial contractions in higher dimensions or to flips, where the exceptional divisor is replaced by a birational transform; nothing in the paper claims this, but the machinery is not threefold-specific.","For singular contracted curves, irreducibility of the exceptional set fails in the current setup, so a different $v$-basis or further tilts will be needed; the paper says the algorithm does not work verbatim in that case."],"forward_implications":["The tables give an explicit finite list of normal forms for divisorial contractions to smooth curves in terminal threefolds, so checking terminality of the proper transform is reduced to checking algebraic conditions on the defining equation.","Every weighted blow-up in the list has ambient dimension at most 5 and an exceptional divisor whose non-Gorenstein loci are explicitly described as cyclic quotient or cA, cD, cE singularities.","The general elephant conjecture holds for all contractions covered by Theorem 1.2: some surface through the curve has at worst Du Val singularities.","Because the construction is stepwise and algebraic, it supplies a blueprint for extending the classification to singular contracted curves, a case the authors leave to later work."],"supporting_citations":[{"why":"Establishes the smooth base case: a divisorial contraction to a curve from a smooth threefold is the usual blow-up of a smooth curve, which the paper extends.","marker":"[Mor82]"},{"why":"Generalizes the smooth result to Gorenstein threefolds by showing the contraction is the blow-up of an lci curve, a normalization used in the classification.","marker":"[Cut88]"},{"why":"Supplies the explicit description of weighted blow-ups and terminal quotient singularities used throughout the tilting algorithm, including Lemma 2.6.","marker":"[Hay99]"},{"why":"Provides the discrepancy formula and canonical-cover setup used in Lemma 3.17 to pass from $X$ to its cover.","marker":"[KM98]"},{"why":"Its Proposition 2.14 is cited to show canonicity is preserved under the finite cover in Lemma 3.17.","marker":"[DL15]"},{"why":"Theorem 17.10 is cited to obtain the birational morphism extracting the divisor over the canonical cover in Lemma 3.18.","marker":"[Ke92]"},{"why":"Proved the general elephant conjecture for contractions with irreducible fibers, the comparison point for Theorem 1.3.","marker":"[KM92]"}],"fun_headline_variants":["All divisorial contractions to curves are weighted blow-ups","Classification of smooth-curve contractions via weighted blow-ups","Every terminal threefold curve contraction is a weighted blow-up","Smooth-curve contractions classified: weighted blow-ups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The non-Gorenstein part of the main theorem is proved through a proposition that assumes the two threefolds are Q-factorial, while the theorem itself does not state this condition; if Q-factoriality is not automatically present in these contractions, the theorem as stated needs an extra assumption.","fun_headline_variants_meta":{"raw":{"variants":["All divisorial contractions to curves are weighted blow-ups","Classification of smooth-curve contractions via weighted blow-ups","Every terminal threefold curve contraction is a weighted blow-up","Smooth-curve contractions classified: weighted blow-ups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000606,"raw_usage":{"total_tokens":2756,"prompt_tokens":810,"completion_tokens":1946,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":426,"completion_tokens_details":{"reasoning_tokens":1882}},"tokens_in":426,"tokens_out":1946,"duration_ms":14493,"temperature":1.0,"reasoning_tokens":1882,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:30:02.352576+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a divisorial contraction to a smooth curve whose target is a terminal non-Gorenstein threefold that is not Q-factorial and check whether its exceptional valuation can be realized by the tilting construction; if no embedding and weights exist, Theorem 1.1 as stated is false. Alternatively, check any row of the classification tables by computing the weighted blow-up and testing whether the proper transform has only terminal singularities, since a single non-terminal row would break the classification.","supporting_citations":[],"review_version":1}