{"id":"5174c42c-dc03-4171-b3be-012989e6a431","arxiv_id":"2504.14141","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Numerically trivial divisors on families extend after a weak semi-stable reduction when the base is a curve, but fail to extend in base dimensions at least two.","lead":"This paper asks when a divisor that is numerically trivial on an open part of a family can extend to the whole family. It proves the extension always works after a modification when the base is a curve, and gives counterexamples when the base has dimension at least two.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 5.1 depends on an unproved monodromy-splitting statement: without a proof that a cyclic étale cover trivializes Pic^0_{U_C/C}, the nonconstant-map contradiction is incomplete.","rationale":"I read the paper in good faith and focused on what would have to be true for the central claims. The dimension-one extension theorem, Theorem 1.2, is argued through relative Picard schemes, Néron models, and the open immersion Pic^0_{X/S}→A; apart from routine missing details (including the unstated conclusion L≡_S 0 in the theorem statement), the Neron-model argument is coherent and I did not find a fatal flaw in it. The higher-dimensional counterexamples, however, depend on Example 5.1's claim that Pic^0_{U_C/C} becomes split after a cyclic étale cover. This claim is asserted without proof, and the contradiction genuinely uses the split projection to G_m. The reader's weakest_assumption identified exactly this point, and I agree that it is the single most load-bearing concern. It is not a contradiction or a sign of error, but a substantive omitted proof; if the monodromy computation were written out, the proof would likely go through. Therefore the appropriate assessment remains conditional on that repair, and I see no reason to change the reader's verdict.","tokens_in":14794,"tokens_out":32022,"duration_ms":319473,"concrete_test":"Compute the monodromy representation associated with Pic^0_{U_C/C} in Example 5.1: take a loop in C' around the type-III point and compute its action on the character group of the rank-one torus, equivalently on H^1(D(U_t),Z)≅Z of a nearby fiber. If the action is the nontrivial element of Aut(G_m,1)≅Z/2, then the cyclic double cover C''→C' kills it; then explicitly construct the isomorphism Pic^0_{U_{C''}/C''} ≅ G_m×C'' using the character induced by this H^1(D(U_t),Z). Finally verify that the base-changed section α_{m,C''} takes a non-identity value at a type-II point and the identity at a type-V point, preserving the contradiction. If instead the monodromy computation gives trivial monodromy, the splitting is immediate; if the values coincide after pullback, the counterexample fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is in Example 5.1. After base-changing to a curve C dominating the closure C' of the type-(II) locus, the proof asserts that since every fiber of Pic^0_{U_C/C} is G_m, the group scheme is a G_m-bundle with identity section, and that 'possibly after a cyclic etale cover C''→C to kill the monodromy we may assume that Pic^0_{U_C/C} ≃ G_m × C'. No proof or computation is supplied. The subsequent contradiction composes a section C→Pic^0 with the first projection to G_m; if the rank-one torus is non-split, no global projection to G_m exists, and a section of a non-split torus over a complete curve need not be constant in any obvious way. Thus the nonconstancy argument—the heart of the higher-dimensional counterexample leading to Theorems 1.3 and 1.4—rests entirely on this unproved splitting. The assertion is plausible: a rank-one torus over a curve has monodromy a homomorphism π1(C)→Z/2, killed by a double cover. But proving it requires a concrete analysis of the relative Picard scheme along C and its dual-graph monodromy, and that analysis is absent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the extension problem for relatively numerically trivial divisors in a projective family f:X→S. The main positive result (Theorem 1.2) states that when S is a smooth curve and f is semi-stable, a Cartier divisor L_U on X_U that is numerically trivial over U extends, up to a uniform integer multiple, to a Q-divisor L on X that is integral after multiplication by m. The proof uses the relative Picard scheme, Néron models, and semi-stable reduction. The paper then gives a counterexample in dimension at least two: over a base mapping to the moduli stack of stable genus-one curves with two marked points, the line bundle formed by the difference of the two marked sections is numerically trivial on an open set but admits no extension to a relatively numerically trivial divisor, even after birational modifications of the family. This yields Theorems 1.3 and 1.4, including a failure of extension for relatively nef divisors.","tokens_in":15000,"tokens_out":15843,"duration_ms":150134,"significance":"If the results are correct, the paper settles a natural question in birational geometry: relatively numerically trivial divisors extend after semi-stable reduction when the base is a curve, but not in higher dimensions. The main positive theorem is proved with standard, well-chosen machinery (Picard schemes, Néron models, [BLR90] and [Kol23]), and the proof is largely coherent. The counterexample is conceptually interesting and, if fully justified, sharp. The use of the moduli space of pointed stable genus-one curves is appropriate. However, the manuscript has two load-bearing gaps: the statement of Theorem 1.2 omits the central conclusion that the extended divisor is relatively numerically trivial, and Example 5.1 relies on an unproved statement about the relative Picard scheme being a split G_m-bundle after an étale cover. These issues are fixable but require substantive additions.","major_comments":[{"comment":"The theorem as printed does not state that the extended Q-divisor L is relatively numerically trivial over S. The abstract and introduction both promise a global divisor L with L ≡_f 0, and the proof indeed establishes this property in the final paragraph, but the statement in Theorem 1.2 lists only L|X_U = L_U and mL integral Cartier. The statement must be corrected to include L ≡_S 0; otherwise the theorem is weaker than advertised and does not match the proof.","section":"§1, Theorem 1.2"},{"comment":"The claim that Pic^0_{U_C/C} is a G_m-bundle with identity section and that, possibly after a cyclic étale cover, Pic^0_{U_C/C} ≃ G_m × C is asserted without proof. This assertion is load-bearing because the contradiction uses the projection onto the first factor to produce a nonconstant morphism from a proper curve to G_m. The text does not even justify that the group scheme is a torus (as opposed to a more general smooth group scheme with G_m fibers). The authors should supply a proof: either compute the monodromy of the associated rank-one torus and show it is killed by a finite étale cover, or replace the splitting step by the observation that a torus over a curve is affine, so any morphism from a proper connected curve to it is constant. The latter would make the splitting argument unnecessary and would also avoid the missing computation.","section":"§5, Example 5.1"},{"comment":"The proof asserts that a nef divisor on the compactification whose restriction to the open locus S is trivial must be an effective vertical divisor and hence a pullback of a divisor on the base. This step is not justified. A nef divisor with zero restriction to an open set does not automatically have an effective representative, and the claim that nefness forces it to be pulled back from S requires an argument (for example, using relative numerical triviality on fibers over S). The current wording is too quick and leaves the proof of Theorem 1.4 incomplete.","section":"§5, proof of Theorem 1.4"}],"minor_comments":[{"comment":"The title contains a typo: 'A F AMILY' should be 'A FAMILY'. The abstract also uses a nonstandard hyphenation of 'numerically' in the title.","section":"Title and abstract"},{"comment":"The text says 'Pic^0_{X/S} is closed in Pic^0_{X/S}' but should say 'closed in Pic_{X/S}'. This typo could confuse readers.","section":"§3, Corollary 3.10(2)"},{"comment":"The notation for the identity element of G_m is inconsistent: the paper writes 'α_m,T(t) = 0' for the trivial divisor class, but when identifying the fiber with G_m the identity should be written as 1 (or 'trivial'). This is presentation only.","section":"§5, Example 5.1"},{"comment":"The lemma states that deg(g)·g_*L' is Cartier for a finite morphism between normal varieties, but for an arbitrary finite morphism (not necessarily flat) the norm of a line bundle is not automatically a line bundle. The applications in the paper are to finite base changes over a smooth curve, which are flat, but the lemma as stated is too general and should either be restricted or supplied with a reference for the non-flat case.","section":"§2, Lemma 2.3"},{"comment":"The commutative diagram labeled 'semi-stable reduction' is typeset in a way that is hard to read; the text is still understandable, but the diagram should be cleaned up for publication.","section":"§4, proof of Theorem 1.2"}],"recommendation":"major_revision","confidential_remarks":"The paper contains several typos and minor notational inconsistencies that need correction. More importantly, the proof of Theorem 1.2 is missing its central statement, and Example 5.1 has an unproved but plausible splitting assertion. Both are fixable within the manuscript's scope, so I do not recommend rejection, but the revision must be substantive rather than editorial."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Xie paper on extending numerically trivial divisors. The headline: the dimension-one positive result is real, and the higher-dimensional counterexample is clever. The paper deserves a serious referee; the main gaps are typos and one unproved splitting assertion that is likely repairable.\n\nWhat's new: Theorem 1.2 gives a clean answer for semistable projective families over a smooth curve: after a multiplication, an open-relative numerically trivial Cartier divisor extends to a Q-divisor that is numerically trivial over the whole base. The proof via Pic^0 and Néron models is the right machinery, and the use of Proposition 3.11 to identify the torus part via the dual complex is a useful technique. The counterexample based on M_{1,2} is also well chosen: the type-(III) stratum is precisely where Σ1−Σ2 stops being numerically trivial, and the argument that no compactification can carry a relatively numerically trivial extension is convincing in spirit.\n\nThe soft spots are not fatal, but they are real. First, Theorem 1.2 as printed does not state that the extended divisor is relatively numerically trivial; that is the whole point, and the proof does deliver it, so it is a fixable omission. Second, Example 5.1 asserts that after a cyclic étale cover Pic^0_{U_C/C} ≅ G_m × C, with no proof. For a rank-one torus over a curve the claim is plausible, but it is not immediate, and the text should either prove it or, better, avoid it: the contradiction only needs the fact that Pic^0 is affine over C, so any regular section from the proper curve is constant. Third, the proof of Theorem 1.3 says ‘any non-empty open subset U’, which cannot be right; the claim only makes sense for the open locus where L is numerically trivial. That is a typo, but it could confuse a reader. Minor computational details (e.g., the multipliers in Theorem 1.4) also want a once-over.\n\nOverall, the math is sound in its main lines and the paper is an honest piece of work: no circularity, no hand-waving about the literature. I would send it to a knowledgeable referee; with a careful revision addressing those points, it should be publishable. For a reading group, it is a good example of Picard-scheme and Néron-model techniques applied to a natural question.","headline":"A solid dimension-one extension theorem plus a clever higher-dimensional counterexample; fix the statement of Theorem 1.2 and the unproved torus splitting.","tokens_in":15544,"tokens_out":9445,"would_cite":true,"duration_ms":91243,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C20","14D06","14K30","14H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A divisor that is numerically trivial over an open subset of a one-dimensional base always extends, after a semi-stable reduction and multiplication by a fixed integer, to a divisor numerically trivial over the whole base; in higher…","keywords":["numerically trivial divisor","relative nef divisor","Picard scheme","Néron model","semi-stable reduction","moduli of stable curves","extension of divisors"],"falsifier":"On the base-changed curve $C$, compute the rational function $\\mathrm{pr}_1\\circ\\alpha_{m,C}$ on the dense open where it is defined and inspect its behavior at a point of type (III); the counterexample requires this function to have a pole there, since a regular map from a proper curve to $\\mathbb{G}_m$ must be constant.","tokens_in":14533,"feed_emoji":"📐","tokens_out":11783,"duration_ms":100109,"temperature":0.7,"pith_summary":"This paper asks whether a divisor on the fibers of a family that is numerically trivial—meaning it has degree zero on every curve in every fiber—over an open subset of the base can be extended to the whole family, possibly after modifying the family. The answer is yes when the base is a one-dimensional curve: after a semi-stable reduction, some fixed multiple of the divisor, with the multiple depending only on the family, extends to an integral Cartier divisor that is numerically trivial over every fiber. The answer is no in general for bases of dimension two or more. The counterexample is built from the moduli stack of stable genus-one curves with two marked points, where the divisor given by the difference of the two marked points is numerically trivial on an open locus but cannot be extended to any birational model of the family, and the same failure afflicts relative nef divisors.","feed_headline":"Numerically trivial divisors extend on curve bases","feed_subtitle":"A fixed multiple extends after semi-stable reduction; bases of dimension at least two can fail.","key_machinery":"The load-bearing object is the relative Picard scheme $\\mathrm{Pic}_{X/S}$, whose identity component $\\mathrm{Pic}^0_{X/S}$ parameterizes line bundles on fibers that are algebraically equivalent to zero; numerically trivial line bundles land in a fixed multiple of this component by Lemma 3.7. The proof compares $\\mathrm{Pic}^0_{X/S}$ with the Néron model of its generic fiber—the canonical smooth group scheme over the base through which every rational section extends uniquely—and multiplies by an integer $m$ that kills the finite component groups of the special fibers. For simple normal crossing fibers, Proposition 3.11 identifies $\\mathrm{Pic}^0$ with a semi-abelian variety whose torus part is governed by the first homology of the dual complex of the fiber. The counterexample uses the same Picard scheme over a curve: fiberwise $\\mathrm{Pic}^0$ is $\\mathbb{G}_m$, so the scheme is a $\\mathbb{G}_m$-bundle, and the section coming from the divisor cannot extend because a regular map from a proper curve to $\\mathbb{G}_m$ is constant.","core_discovery":"The paper proves Theorem 1.2: if $f:X\\to S$ is a semi-stable projective family over a smooth curve, $U\\subset S$ is open, and $L_U$ is a Cartier divisor on $X_U$ with $L_U\\equiv_U 0$, then there is an integer $m>0$, depending only on $f$, and a $\\mathbb{Q}$-divisor $L$ on $X$ such that $L|_{X_U}=L_U$, $mL$ is an integral Cartier divisor, and $L$ is relatively numerically trivial over $S$. The paper also proves that this is sharp in dimension: there is a projective semi-stable contraction $f:X\\to S$ with $\\dim S\\ge 2$ and a Cartier divisor $L$ on $X$ such that $L$ is numerically trivial over an open subset $U\\subset S$, yet no $\\mathbb{Q}$-divisor extending $L|_U$ is numerically trivial over $S$, even after arbitrary birational modifications of the family. The same example shows that a divisor nef over an open subset need not extend to a nef divisor over any compactification of the family.","pith_inferences":["The mechanism suggests that the real obstruction is the monodromy of $\\mathrm{Pic}^0$: whenever $\\mathrm{Pic}^0_{X/S}$ is proper over the base, generic sections should extend without a multiplication step, so the extension problem reduces to controlling torus parts in special fibers.","A natural testable extension is to search for a curve-base analogue of Example 5.1 in which $\\mathrm{Pic}^0$ is a non-split $\\mathbb{G}_m$-torsor; the paper's open Question 1.5 would be answered negatively if such a family also blocks nef extension.","Because the paper works over $\\mathbb{C}$, the Picard-scheme and Néron-model arguments suggest the one-dimensional extension theorem may survive in positive or mixed characteristic, provided the cited representability and smoothness statements hold there."],"forward_implications":["For any semi-stable family over a smooth curve, the extension obstruction for numerically trivial divisors is annihilated by one fixed multiple $m$ depending only on the family, so the same $m$ works for every divisor and every open subset.","The same conclusion holds for locally stable families over a DVR with perfect residue field, where the full semi-stable reduction is not required.","For bases of dimension at least two, the paper's example shows that even after replacing the total space by any higher birational model, no nonzero multiple of a relatively numerically trivial divisor on an open subset extends.","The counterexample also rules out the nef version of extension in dimension at least two: a divisor nef over an open subset can fail to extend to a nef divisor over any compactification of the family.","The one-dimensional nef version remains open, and the paper records that it is equivalent to the existence of a Zariski decomposition of the divisor on higher models."],"supporting_citations":[{"why":"Supplies the Néron model of an abelian variety over a Dedekind scheme and the mapping property used to extend a generic section after multiplication.","marker":"[BLR90]"},{"why":"Provides existence, separatedness, and basic structure of Picard schemes, including the identity component $\\mathrm{Pic}^0$.","marker":"[Kl05]"},{"why":"Supplies semi-stable reduction, used to reduce to a family whose fibers are simple normal crossing and admit a section.","marker":"[KKMS73]"},{"why":"Gives the descent result used in Lemma 2.2 to push a numerically trivial divisor down through rational singularities.","marker":"[KM92]"},{"why":"Supplies the local-stability, slc-fiber, and cohomology-flatness facts used to construct the open group subscheme $\\mathrm{Pic}^0_{X/S}$.","marker":"[Kol23]"},{"why":"Provides the rational chain connectedness used in Lemma 3.9 to show generic sections extend when $\\mathrm{Pic}^0_{X/S}$ is proper.","marker":"[HM07]"}],"fun_headline_variants":["Curve bases extend numerically trivial divisors","Higher-dimensional bases block divisor extension","Extension works on curves, fails in higher dimensions","Semi-stable reduction enables curve-base extension"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The higher-dimensional counterexample rests on the unproved assertion that after a cyclic étale cover the relative Picard scheme is a product $\\mathbb{G}_m\\times C$, because if the monodromy could not be killed by such a cover, the rational section would live on a non-split $\\mathbb{G}_m$-torsor and the contradiction would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Curve bases extend numerically trivial divisors","Higher-dimensional bases block divisor extension","Extension works on curves, fails in higher dimensions","Semi-stable reduction enables curve-base extension"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000902,"raw_usage":{"total_tokens":3897,"prompt_tokens":977,"completion_tokens":2920,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":2866}},"tokens_in":593,"tokens_out":2920,"duration_ms":21000,"temperature":1.0,"reasoning_tokens":2866,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:56:21.868430+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the base-changed curve $C$, compute the rational function $\\mathrm{pr}_1\\circ\\alpha_{m,C}$ on the dense open where it is defined and inspect its behavior at a point of type (III); the counterexample requires this function to have a pole there, since a regular map from a proper curve to $\\mathbb{G}_m$ must be constant.","supporting_citations":[],"review_version":1}