{"id":"cb295e1f-2c97-4264-9591-d78485e697fa","arxiv_id":"2607.07053","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On very general abelian varieties of dimension ≥4 (and genus-4 Jacobians), D^2 = 0 in CH^2 forces D torsion; consequently all rational sections of the genus-4 Kummer fibration are rational multiples of the Griffiths–Pirola section.","lead":"This paper proves that on very general abelian varieties of dimension at least four, a divisor whose square vanishes in the Chow group must be torsion, and that the same is true for very general genus-4 Jacobians. It then uses the Jacobian case to prove Pirola's conjecture: all rational sections of the genus-4 Kummer fibration are multiples of the Griffiths–Pirola section.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.8 rests on [20, Thm 1.4] (CH_0(C^(2)_η) one-dimensional, generated by γ_P^2), which is not proved here; if that companion result fails, the Pirola-conjecture proof collapses.","rationale":"The reader's weakest_assumption already identified Proposition 4.2's reliance on [20, Theorem 1.4] as the most fragile point. I agree: Theorem 1.8, the paper's second headline, cannot get off the ground without the 1-dimensionality statement for CH_0(C^(2)_η)_hom ⊗ Q. This is not an internal inconsistency — the paper is explicit that the result comes from a companion — but it is a load-bearing external dependency. The other fragile input, Ax–Schanuel [6, Theorem A] in Proposition 3.16, is also important; however Proposition 3.16 supports the proof of Theorem 1.2(ii), and even if one accepted it, the Pirola conjecture still depends on the companion's structural result. Thus the companion dependency is the single most consequential unchecked premise. I saw no evidence of circularity or error in the rest of the argument; the apparent factor-2 descent formula in Section 4 is at most a typo, since only proportionality is used. Given the external theorem is citable and apparently proved independently in [20], the appropriate verdict remains ACCEPT/UNCHANGED rather than REJECT, pending the concrete check above.","tokens_in":28491,"tokens_out":31700,"duration_ms":281283,"concrete_test":"Obtain companion arXiv:2607.12793 and verify [20, Theorem 1.4] line by line: (a) confirm the upper bound CH_0(C^(2)_η)_hom⊗Q ⊆ Q·γ_P^2 is proved for the generic genus-4 curve over M_4, not just for a fixed quadric, and does not already assume the conclusion of Theorem 1.8; (b) recompute the infinitesimal invariant δ(D^2_{γ_P}) from [20, Thm 1.3] to confirm it is nonzero, so the claimed generator is nonzero. If both hold, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.8 proceeds through Proposition 4.2, which asserts D^2_{γ,b} = λ D^2_{γ_P,b} in CH_0(C_b^(2)). The only argument given for Proposition 4.2 is 'Applying Theorem 4.1 to q_*D^2_{γ′}', where Theorem 4.1 = [20, Theorem 1.4] is quoted from the companion preprint: CH_0(C^(2)_η)_hom ⊗ Q is 1-dimensional and generated by γ_P,η^2. The present paper proves neither the upper bound (attributed to [21]/[12]) nor the nonvanishing (from [20, Thm 2.10]). This is not circular — the text says the companion proof is direct — but it is a genuine structural dependency: if [20, Thm 1.4] is wrong or applies only to complete intersections in a fixed quadric rather than to the universal family over M_4, then Eq. (65) fails and the descent to Corollary 4.4 and Proposition 4.5 has no starting point. The companion is not included in this submission, so the reader cannot check whether it secretly uses Theorem 1.2 or whether the generator identification with γ_P^2 is correct for the generic genus-4 curve over M_4.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two main theorems. Theorem 1.2 states that on a very general abelian variety of dimension at least 4, any divisor D whose square vanishes in CH^2 is torsion, and the analogous statement holds for a very general genus-4 Jacobian. Theorem 1.8 resolves a conjecture of Pirola: every rational section of the Kummer fibration K=J/±Id over M_4 is a rational multiple of the Griffiths-Pirola section. The proof of Theorem 1.2 combines infinitesimal invariants of normal functions with an algebraic rank-1 reduction (Proposition 3.1), an integrability/horizontality argument (Proposition 3.10), and an Ax-Schanuel input (Proposition 3.16). The proof of Theorem 1.8 uses the one-dimensionality of CH_0 of the generic second symmetric product from the companion paper [20] to compare the square of an arbitrary section with the Griffiths-Pirola square, then applies the infinitesimal invariant analysis of Section 4.1.","tokens_in":28826,"tokens_out":9938,"duration_ms":97171,"significance":"If correct, these are substantial results. Theorem 1.2 gives the first complete description of zero-square divisors on very general abelian varieties of dimension at least 4, and the genus-4 Jacobian case is used for the Pirola conjecture. Theorem 1.8 is a strong Franchetta-type statement for the Kummer fibration in genus 4 and goes beyond previously known results. The paper contains detailed and original algebraic work, especially Proposition 3.1, Claim 3.13, Proposition 4.5, and the monodromy arguments. The main theorems are falsifiable and the structure of the proof is coherent. However, as noted below, several load-bearing ingredients are quoted from the companion preprint [20], and the present manuscript does not include proofs of those ingredients; this makes independent verification difficult and is the main reason the paper needs revision before acceptance.","major_comments":[{"comment":"Theorem 1.8 depends crucially on [20, Theorem 1.4], quoted as Theorem 4.1: CH_0(C^(2)_η)_hom⊗Q is 1-dimensional and generated by γ_P,η^2. This is the only input that yields Eq. (65), D^2_{γ,b}=λD^2_{γ_P,b}, and hence Corollary 4.4 and Proposition 4.5. The present paper proves neither the upper bound (from [21]/[12]) nor the nonvanishing (from [20, Thm 2.10]). The companion is not included in this submission, so the referee cannot check whether the generator identification is correct for the generic genus-4 curve over M_4, nor whether the companion secretly uses Theorem 1.2. I ask that the companion proof be included in the submission, or that the dependence on [20, Thm 1.4] be removed or made verifiable.","section":"Section 4, Proposition 4.2 and Theorem 4.1"},{"comment":"The identification of the deepest piece L^2H^2(Ω^2_{J_g|J_{g,m}}) with I_{2,W} (resp. I_{2,V}), and the statement that the square map is M_{2,W} (resp. M_{2,V}), are quoted from [20] (including [20, Prop. 5.1]). This identification is load-bearing for Theorem 1.2, because it converts the vanishing δ(D^2)=0 into the algebraic equation M_{2,W}(δD)=0 that is subsequently solved in Proposition 3.1. Without a proof or an exact statement from the companion, the algebraic core of Theorem 1.2 is not self-contained. Please provide the missing proof or include the companion as an appendix.","section":"Section 3.1, equations (18)–(22) and following text"},{"comment":"The application of the Ax-Schanuel theorem [6, Theorem A] is very compressed. In particular: (i) the hypotheses of [6, Thm A], which concern special subvarieties and leaves of a connection on a principal bundle, need to be checked for the frame bundle of a generically finite cover M→A_g (resp. M→M_g); (ii) the statement that the images p(F_e∩F_L) are contained in proper ∇-special subvarieties is asserted in one sentence and is not immediate, because F_e∩F_L is an intersection inside the frame bundle and its projection is not obviously special; (iii) the passage from this to algebraic integrability of the distribution α requires explanation. This is a correctness-risk concern. I would recommend expanding this argument, even if the statement is ultimately correct.","section":"Section 3.2, Proposition 3.16"}],"minor_comments":[{"comment":"Typo: 'neeeded' should be 'needed'.","section":"Introduction, after Corollary 1.3"},{"comment":"In the paragraph after Eq. (8), 'η_F vanishes identically' should presumably read 'η^{1,0} vanishes identically'.","section":"Section 2, proof of Lemma 2.1"},{"comment":"The notation 'deepest part L^2H^2(Ω^2...)' is used without a precise definition of the filtration degree indexing; this can be confusing because L^* has a decreasing indexing in the preceding subsection. Please clarify whether L^2 denotes the second step of the filtration or its graded piece.","section":"Section 3.1, before Eq. (22)"},{"comment":"The line 'On any irreducible component of this double cover, we have by Theorem 4.9 that γ'=λγ_P' uses a rational λ, while Theorem 4.9 is stated with integers N,N'. The passage from Nγ=N'γ_P to a rational coefficient should be made explicit, including the treatment of components where γ' is torsion.","section":"Section 4.2, Lemma 4.10"},{"comment":"The spelling of Roitman is inconsistent: [7] uses 'Roitman' and [18] uses 'Rojtman'. Please standardize.","section":"References"},{"comment":"Remark 1.5 says the higher-genus extension needs 'Proposition 3.1(ii)' in higher genus; this should really be a higher-genus analogue of Proposition 3.1(ii), since the current statement is for g=4. The wording is ambiguous.","section":"Remark 1.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct and important, but the reliance on the unpublished companion [20] is a genuine obstacle to verification. If the companion can be made available and its proofs checked, I would be willing to accept. The Ax-Schanuel application also needs expansion, though I do not see an obvious error. I would suggest the editor obtain [20] before making a final decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here’s the short version: this paper proves that for a very general abelian variety of dimension at least 4, any divisor D with D^2 = 0 in CH^2 is torsion, and uses the genus-4 Jacobian case to prove Pirola's conjecture about rational sections of the Kummer fibration. That is a real result, not a repackaging.\n\nWhere credit is due: the proof of Theorem 1.2 is mostly self-contained, and the algebraic heart—Proposition 3.1, the rank-one lift analysis—is written out in full. The paper is also honest about what it imports. The imported items are genuine citations: [20] for the infinitesimal square map, [6] for Ax–Schanuel, and the author's earlier [23] for the dimension-zero locus. The new statement is torsion, not just dimension zero, so this is a genuine extension.\n\nThe soft spot is the one the skeptic flagged. Theorem 1.8's starting point, Proposition 4.2, relies on [20, Theorem 1.4]—the one-dimensionality of CH_0(C^(2)_η) generated by γ_P^2—which is not proved here. The upper bound comes from [21] and nonvanishing from [20]. If that companion result fails, or does not cover the universal genus-4 curve via the (2,3) complete intersection family, then equation (65) has no starting point. This is a structural dependency, not a circular one—the paper does not assume the torsion theorem to get it—but Theorem 1.8 cannot be independently certified from this manuscript alone. That is worth stating clearly, though it is a reason to send both papers to the same referee, not to desk-reject.\n\nA second external input sits at the core of Proposition 3.16: the use of [6, Theorem A] about Ax–Schanuel and special subvarieties. The setup looks standard and the argument is plausible, but a referee should check that the leaf codimension and monodromy density hypotheses are satisfied in exactly this setting. I would not call it fragile, but it is another point where the paper is leaning on an outside theorem in a slightly delicate way.\n\nWho this paper is for: algebraic geometers working on Chow rings, normal functions, and the Franchetta–Pirola circle. It deserves a serious referee. I would send it out, together with the companion preprint, and expect heavy but likely successful revision.","headline":"A serious paper proving a long-open torsion statement for very general abelian varieties and Pirola's conjecture, with a real but standard structural reliance on a companion preprint.","tokens_in":29338,"tokens_out":2315,"would_cite":true,"duration_ms":23442,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C15","14D07","14H40","14K12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that on very general abelian varieties of dimension at least 4, every divisor whose square vanishes in the Chow ring is torsion, and uses this to show that all rational sections of the genus-4 Kummer fibration are multiples","keywords":["Chow ring","abelian varieties","zero-square divisors","torsion divisors","infinitesimal invariants","normal functions","Kummer fibration","genus 4 Jacobians"],"falsifier":"Produce a very general abelian fourfold (or a very general genus-4 Jacobian) with a non-torsion divisor D in Pic^0 satisfying D^2 = 0 in CH^2; this would directly falsify the paper's main theorem. Alternatively, exhibit a rational section of the genus-4 Kummer fibration that is not a multiple of the Griffiths-Pirola section.","tokens_in":28350,"feed_emoji":"📐","tokens_out":8915,"duration_ms":74749,"temperature":0.7,"pith_summary":"This paper tries to settle two related questions in the Chow theory of abelian varieties. First, it argues that on a very general abelian variety of dimension at least four, any divisor whose square vanishes in the Chow ring must be torsion; the same holds for the Jacobian of a very general genus-four curve. Second, it proves that every rational section of the Kummer fibration attached to the universal genus-four Jacobian is a rational multiple of one special section, the Griffiths-Pirola normal function. These results matter because they sharply constrain the ring structure of Chow groups and connect it to transcendence statements of Ax–Schanuel type.","feed_headline":"Zero-square divisors on generic abelian varieties are torsion","feed_subtitle":"Settles the genus-4 Kummer fibration: only multiples of one section occur.","key_machinery":"The central machinery is the Griffiths infinitesimal invariant of a normal function, a first-order derivative measuring how a family of cycles moves in a family of abelian varieties, together with its extension to codimension-2 cycles. The key algebraic step is the square map M_2, which acts as the 2-by-2 minors map on tensors. Proposition 3.1 shows that M_2(ϕ)=0 forces a lift of ϕ to be rank-one. That rank-one property feeds into Proposition 3.16, which uses the Ax–Schanuel theorem to show that nonzero algebraic flat locally decomposable sections of H^1⊗Ω_M do not exist here. For the section theorem, the same square map is used to solve M_2(ϕ)=λ M_2(δγ_P), whose only solutions are multiples","core_discovery":"The paper proves two theorems. First, for a very general abelian variety of dimension at least 4, a divisor D in Pic^0(A) with D^2 = 0 in CH^2(A) must be torsion; the same is true for the Jacobian of a very general genus-4 curve. Second, all rational sections of the Kummer fibration of the universal genus-4 Jacobian are rational multiples of the Griffiths-Pirola normal function, the section defined by the difference of the two trigonal divisors. The proof uses Griffiths infinitesimal invariants of normal functions; the vanishing of the square forces the invariant to have a rank-one representative, and an Ax–Schanuel-type theorem rules out nonzero algebraic flat locally decomposable sections,","pith_inferences":["If the same infinitesimal-invariant strategy can be extended to higher powers, one would expect that on very general abelian varieties of dimension at least 2k, the condition D^k = 0 in CH^k(A) forces D to be torsion; that would be a natural test of the generality of the rank-one mechanism.","The role of Ax–Schanuel suggests a general principle: algebraic flat sections of H^1 ⊗ Ω_M that are locally decomposable are forced to vanish whenever the monodromy group is large; families with small monodromy could provide counterexamples to the zero-square statement.","The integer-multiple conclusion for the genus-4 section theorem hints that normal-function multipliers in moduli problems may often be integral; this could be explored in other Franchetta-type settings.","The paper explicitly acknowledges that the genus-higher version of the second symmetric product computation is not done here; extending the relevant result to genus greater than 4 is a concrete next step."],"forward_implications":["On very general abelian varieties of dimension at least 4, the Chow ring has no exotic zero-square divisors: the set of D in Pic^0 with D^2 = 0 is exactly the torsion subgroup.","The same conclusion holds for very general genus-4 Jacobians, and equivalently for divisors on their second symmetric products.","The only rational sections of the genus-4 Kummer fibration are multiples of the Griffiths-Pirola section; in particular, no new independent section appears.","The paper conjectures that on very general abelian varieties of dimension at least 2k, the only divisors with D^k = 0 are torsion."],"fun_headline_variants":["Torsion ruling for zero-square divisors on generic abelian varieties","Pirola conjecture proven: only Griffiths-Pirola Kummer sections","Zero-square divisors in CH^2 are torsion on generic Jacobians","Generic abelian varieties: zero-square divisors are torsion","Genus-4 Jacobians: zero-square divisors force Pirola result"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The classification of rational sections of the genus-4 Kummer fibration relies on a cited result from the companion paper — that the space of zero-cycles modulo rational equivalence on the generic double symmetric product of a genus-4 curve is one-dimensional, generated by the square of the Griffiths-Pirola divisor — which is not proved in the present paper.","fun_headline_variants_meta":{"raw":{"variants":["Torsion ruling for zero-square divisors on generic abelian varieties","Pirola conjecture proven: only Griffiths-Pirola Kummer sections","Zero-square divisors in CH^2 are torsion on generic Jacobians","Generic abelian varieties: zero-square divisors are torsion","Genus-4 Jacobians: zero-square divisors force Pirola result"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1383,"prompt_tokens":666,"completion_tokens":717,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":626}},"tokens_in":410,"tokens_out":717,"duration_ms":6687,"temperature":1.0,"reasoning_tokens":626,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T08:06:45.494935+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce a very general abelian fourfold (or a very general genus-4 Jacobian) with a non-torsion divisor D in Pic^0 satisfying D^2 = 0 in CH^2; this would directly falsify the paper's main theorem. Alternatively, exhibit a rational section of the genus-4 Kummer fibration that is not a multiple of the Griffiths-Pirola section.","supporting_citations":[],"review_version":2}