{"id":"8eb4471a-41ab-4edd-82b8-1b81981ba733","arxiv_id":"2608.12950","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A very general prime Fano threefold of genus 7 admits no multiplicative Chow-Künneth decomposition, because the explicit cycle Z_Y on Y×Y is Abel-Jacobi trivial but non-zero in the Chow group.","lead":"For the very general prime Fano threefold of genus 7, the author constructs an explicit 2-cycle on Y×Y that is non-zero in the Chow group despite being trivial in cohomology. This gives the first known negative answer to a question about multiplicative Chow-Künneth decompositions for such threefolds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Chow-level motivic isomorphism of Prop. 2.6 is the load-bearing step, but its proof is a sketch whose homological-to-Chow upgrade is not generally valid; if it fails, the transfer maps (4) and the contradiction collapse.","rationale":"The reader's conditional verdict is correct. Our review identifies the same load-bearing assumption (Proposition 2.6) as the reader's weakest_assumption. The non-vanishing of Z_Y, and hence the negative answer to Question 1.3, is entirely mediated by the transfer from the dual curve C via the injective maps (4). Those maps are a consequence of the Chow-level motivic isomorphism and the generic definiteness of its correspondence. The paper offers three proof sketches, but none is complete: the Kimura upgrade is not a theorem in general, and the birational diagram argument depends on an unproved flop diagram and on identifying the two blow-ups. We agree that the verdict should remain CONDITIONAL pending a complete proof of Proposition 2.6 (and the explicit verification that the transfer sends Z_C to a nonzero multiple of Z_Y). We also note that the Abel–Jacobi triviality claim, while part of the theorem statement, is not required for the no-MCK corollary; the latter only needs Z_Y to be homologically trivial and non-zero.","tokens_in":12940,"tokens_out":45154,"duration_ms":419592,"concrete_test":"Verify Proposition 2.6 for a smooth genus-7 example: construct the correspondence Γ from the HPD kernel as in Lemma 2.19, and check that Γ∘Γ^∨ = π^3_Y in CH^3(Y×Y) and Γ^∨∘Γ = π^1_C(−1) in CH^1(C×C), where π^i are the Chow–Künneth projectors. In parallel, check the birational diagram (1) by computing the two blow-ups: confirm that blowing up the 14 lines in \\tilde Y is isomorphic to blowing up the 14 trisecants in \\tilde Q (e.g., compare normal bundles). If the compositions differ from the projectors by a nonzero homologically trivial cycle, or if the two blow-ups are not isomorphic, the Chow-level isomorphism fails and the transfer maps (4) are not justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central argument depends on Proposition 2.6: h(Y) ≅ h(C)(−1) ⊕ 1 ⊕ 1(−3) in M_rat. This Chow-level isomorphism is not established. The first proof option (upgrading an isomorphism in M_hom using Kimura finite-dimensionality) is not a valid general principle: two finite-dimensional motives isomorphic in M_hom need not be isomorphic in M_rat unless the correspondences are rationally trivial modulo homological equivalence. The second option (Bloch–Srinivas) is only a paragraph. The third, via the birational diagram (1), is the most substantive, but it relies on unproved assertions from [17, Thm. 4.4.11(iii)] and [15]/[16], specifically that the flop is realized by blowing up 14 lines in \\tilde Y and 14 trisecants in \\tilde Q, and that the resulting variety Z is the same on both sides. If any of these steps fails, the injective maps (4) GDA^i_B(C×C) → GDA^{i+2}_B(Y×Y) (which require Proposition 2.6 plus Lemma 2.19) are not available, and the contradiction using the Faber–Pandharipande cycle does not go through. The proof of Theorem 3.1 also asserts without computation that (4) sends Z_C to a non-zero multiple of Z_Y, which is needed for the Abel–Jacobi triviality statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies algebraic cycles on very general prime Fano threefolds of genus 7, in the context of Beauville's splitting property conjecture and Shen--Vial's multiplicative Chow--Künneth (MCK) decompositions. The main theorem exhibits an explicit cycle Z_Y in A^4(Y×Y), defined using the diagonal and powers of the anticanonical class, and claims that Z_Y is Abel--Jacobi trivial but nonzero; consequently Y admits no MCK decomposition. The proof passes to the dual genus-7 curve C via a Chow-motive isomorphism h(Y) ≅ h(C)(-1) ⊕ 1 ⊕ 1(-3), then transfers the nontriviality of the Faber--Pandharipande cycle on C×C (Green--Griffiths) to Y×Y. The paper also proves a positive result: every Fano threefold admits an MCK decomposition modulo algebraic equivalence.","tokens_in":13232,"tokens_out":38965,"duration_ms":390506,"significance":"If the main theorem is correct, it gives a negative answer to Question 1.3 for the genus-7 case, complementing the affirmative answers for cubic threefolds, intersections of two quadrics, intersections of a quadric and a cubic, and prime Fano threefolds of genera 8 and 10. The paper also provides one of the few explicit Abel--Jacobi-trivial nonzero cycles on a variety of dimension six. The strategy is attractive and largely reductionist: it uses the classical Green--Griffiths theorem for curves, with the 1/12 coefficient forced by -K_Y^3 = 12. The paper is also careful about the Franchetta property for Y and Y^2, and the mod-algebraic-equivalence statement for all Fano threefolds is a useful complement. The main caveat is that the central Chow-motive isomorphism is presented as a sketch, and the relative version needed for the transfer maps is not fully demonstrated.","major_comments":[{"comment":"Proposition 2.6 is load-bearing, but its proof is only a sketch. The first option (homological isomorphism plus Kimura finite-dimensionality) does not specify the Chow correspondence f whose class is an isomorphism in M_hom; one must explicitly construct f and its inverse and check that the relevant Kimura-finiteness hypothesis applies to the endomorphisms before upgrading from M_hom to M_rat. The sentence 'check that it is induced by a correspondence' is not a proof. The second option is a one-sentence appeal to Bloch--Srinivas. The third option depends on assertions from [17, Thm. 4.4.11(iii)] and [15]/[16] about the flop, the 14 lines and 14 trisecants, and the identification of the common blow-up Z; these are not proved or quoted precisely enough for the role they play. Since all later transfer maps (4) and hence the contradiction in Theorem 3.1 depend on this Chow-level isomorphism, a complete proof or a precise reference with the full statement is required.","section":"§2.5, Proposition 2.6"},{"comment":"The relative version of the Chow-motive isomorphism is not established at the level needed for the transfer maps (4). Lemma 2.19 asserts that the correspondences of Proposition 2.6 are generically defined over B and induce an isomorphism GDA^2_B(Y) ≅ GDA^1_B(C)⊕Q^2, but the passage from the HPD Fourier--Mukai kernel to a relative Chow correspondence is only sketched; it does not justify why the relative correspondence induces an isomorphism of Chow motives fiberwise, as opposed to an isomorphism modulo homological equivalence. In addition, the universal family C→B of dual curves is never formally defined, although the groups GDA^i_B(C×C) in (4) presuppose such a family. Please define C→B and give the relative correspondence explicitly, or provide a precise reference for the relative HPD statement.","section":"§2.5, Lemma 2.19 and §3, Eq. (4)"},{"comment":"In the last paragraph of the proof, the statement that 'the above argument actually shows that the map (4) sends Z_C to a non-zero multiple of Z_Y' is asserted without computation or reference. This is the only step that yields the Abel--Jacobi triviality of Z_Y. One needs either an explicit computation of the image of the Green--Griffiths cycle under the relative correspondence, or a theorem asserting that an isomorphism of Chow motives of the form h(Y) ≅ h(C)(-1)⊕1⊕1(-3) preserves the Abel--Jacobi kernel. As written, the full statement of Theorem 3.1 ('Abel--Jacobi trivial but non-zero') is not justified, even though the non-zeroness may follow from the preceding contradiction argument.","section":"§3, proof of Theorem 3.1"}],"minor_comments":[{"comment":"The conventions paragraph appears to contain a typo: it defines A^j(Y) as the Chow group of j-dimensional cycles, but then says A^j(Y) and A^{n-j}(Y) are used interchangeably, which is the convention for codimension j. The body uses A^j as codimension. Please correct this.","section":"Conventions"},{"comment":"The statement that A^j_hom(Y)=0 for j≠2 is confusing without the codimension convention; A^2_hom(Y) may be nontrivial. The argument only needs the vanishing for the generically defined cycles, so please restate it in the notation used in the paper.","section":"§2.5, proof of Proposition 2.18"},{"comment":"The transition from Z_Y=0 to the Franchetta property for Y^2 is compressed. Please spell out that the relation kills all products of Δ with positive powers of p_i^*H, so that GDA^*_B(Y×Y) reduces to the pullback subalgebra plus Q[Δ] as a vector space; this is what makes the injectivity into cohomology immediate.","section":"§3, proof of Theorem 3.1"},{"comment":"The sentence 'the injective map B^*(C^m)→B^*(Y^m) provided by Proposition 2.6' refers to a statement that is only given for m=1. Please state explicitly that the map for m>1 is obtained by taking tensor products of the correspondence from Proposition 2.6.","section":"§4.2, proof of Corollary 4.5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is clearly within the scope of the journal and fits the author's established program on Chow rings and MCK decompositions. The main risk is the unproved Proposition 2.6 and its relative version, on which the central contradiction depends. I would not recommend rejection on the available evidence, because the strategy is coherent and multiple proof routes are indicated, but the revision should supply a complete proof or precise references for these steps. There is also a minor grading/notation inconsistency in the conventions that should be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Candid take: the paper has a real idea and the main result is likely true, but the proof as written has a hole. What is new: an explicit 2-cycle Z_Y on Y×Y that is shown to be non-zero (modulo the missing piece), giving the first negative answer to Question 1.3 for prime Fano threefolds of genus 7. The strategy is smart: use the dual curve C, the Franchetta property, and the Green–Griffiths cycle. The separate result that any Fano threefold has an MCK decomposition modulo algebraic equivalence also looks solid and is a nice complement.\n\nThe soft spot is central. The proof of Theorem 3.1 never shows that the transfer map (4) sends Z_C to a non-zero multiple of Z_Y. That sentence is the linchpin. Without a computation of the correspondence, the contradiction only gives non-vanishing of some cycle in the image, not of Z_Y itself. The claimed Abel–Jacobi triviality of Z_Y is similarly unsupported; it would follow from the same missing identification but is not derived. The author should either compute the image of Z_C explicitly or state the theorem without the Abel–Jacobi claim.\n\nOn Proposition 2.6, the proof is sketchy, but I disagree with the stress-test's specific objection: for Kimura-finite motives, a homological isomorphism induced by a correspondence does upgrade to a Chow isomorphism, because homologically trivial endomorphisms are nilpotent and so id+h is invertible. That route can work, provided the correspondence realizing the Jacobian isomorphism is actually exhibited. The birational diagram route depends on unproved assertions from the literature and needs more care. The step from Z_Y = 0 to the direct-sum description of GDA^* and the Franchetta property is also compressed.\n\nThe paper deserves peer review. The main result is important if correct, and the missing pieces are likely fillable. I would not cite it in its current form, but I would want to see a revision with the cycle computation.","headline":"Promising negative answer to Question 1.3 for genus 7, but the proof omits the crucial identification between the dual-curve cycle and Z_Y; deserves a serious referee but needs a computation.","tokens_in":658,"tokens_out":1348,"would_cite":false,"duration_ms":121620,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C15","14C25","14C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For very general prime Fano threefolds of genus 7, an explicit cycle on Y×Y is Abel–Jacobi trivial but nonzero in the Chow group, so Y admits no multiplicative Chow–Künneth decomposition.","keywords":["algebraic cycles","Chow groups","multiplicative Chow–Künneth decomposition","Fano threefolds","genus 7","Franchetta property","spinor tenfold","Abel–Jacobi trivial cycles"],"falsifier":"Compute an explicit rational equivalence in $A^4(Y\\times Y)$ for $Z_Y$ on a single very general genus-7 Fano threefold, or prove that the universal family $Y\\times_B Y$ satisfies the Franchetta property in codimension 4; either would directly contradict Theorem 3.1.","tokens_in":12712,"feed_emoji":"","tokens_out":8879,"duration_ms":74378,"temperature":0.7,"pith_summary":"This paper establishes that a very general prime Fano threefold of genus 7 carries an explicit algebraic cycle on its self-product that is invisible to the Abel–Jacobi map yet nonzero in the Chow group. Because such a cycle exists, these varieties do not admit a multiplicative Chow–Künneth decomposition, the current concrete form of the splitting-property conjecture. The proof transfers a known nontrivial zero-cycle from the square of the dual curve to the square of the threefold, and shows the transfer forces the failure of the Franchetta property for Y×Y. The paper also proves that every Fano threefold does admit such a decomposition modulo algebraic equivalence, so the obstruction is finer than algebraic equivalence.","feed_headline":"Explicit 2-cycle is nonzero on genus-7 Fano threefolds","feed_subtitle":"The cycle is Abel-Jacobi trivial yet nontrivial in the Chow group, blocking a multiplicative Chow-Künneth splitting.","key_machinery":"The load-bearing object is the explicit cycle $Z_Y$, a combination of the diagonal class and pullbacks of powers of the polarization. The proof is carried by a chain of equivalences: the Franchetta property for the universal family of genus-7 Fano threefolds, computed via the spinor-tenfold model; a Chow-level isomorphism of motives $h(Y)\\cong h(C)(-1)\\oplus \\mathbf{1}\\oplus \\mathbf{1}(-3)$ relating $Y$ to its dual curve $C$; and generically defined correspondences from homological projective duality that transfer the obstruction. The cycle $Z_C=\\Delta_C\\cdot p_1^*K_C - \\tfrac{1}{12}K_C\\times K_C$ on $C\\times C$ is the known nontrivial input.","core_discovery":"The central result is Theorem 3.1: for a very general prime Fano threefold $Y$ of genus 7 (a smooth Fano threefold whose Picard group is generated by the canonical divisor), with $H=-K_Y\\in A^1(Y)$, the cycle $$Z_Y = \\Delta_Y\\cdot (p_1)^*H - \\tfrac{1}{12}\\big((p_1)^*H\\cdot(p_2)^*$H^{3}$ + (p_1)^*$H^{2}$\\cdot(p_2)^*$H^{2}$ + (p_1)^*$H^{3}$\\cdot(p_2)^*H\\big) \\in $A^{4}$(Y\\times Y)$$ is homologically trivial but nonzero. The argument assumes $Z_Y=0$ and derives that the universal family $Y\\times_B Y$ has the Franchetta property; the Chow-motive isomorphism between $Y$ and its dual curve $C$ then pulls that property back to $C\\times C$, contradicting the known nontriviality of the interesting zero-cycle on $C\\times C$. The contradiction also shows $Z_Y$ is Abel–Jacobi trivial. It follows that $Y$ has no multiplicative Chow–Künneth decomposition.","pith_inferences":["The same transfer strategy may produce explicit nontrivial cycles on other Fano varieties with a dual curve or homological projective duality partner; the paper does not address those cases.","The cycle $Z_Y$ can be viewed as a candidate generator of the codimension-4 Griffiths group of $Y\\times Y$; testing whether it spans that group over a very general fiber would sharpen the statement.","The contrast between Theorem 3.1 and Proposition 4.1 suggests that the right formulation of the splitting-property question for Fano threefolds is modulo algebraic equivalence, where the answer becomes uniformly positive."],"forward_implications":["Very general prime Fano threefolds of genus 7 do not admit a multiplicative Chow–Künneth decomposition.","The universal family $Y\\times_B Y$ fails the Franchetta property in codimension 4, although it holds through codimension 3.","The Chow-motive correspondence between a genus-7 Fano threefold and its dual curve cannot preserve tautological rings even modulo algebraic equivalence.","Every Fano threefold admits a multiplicative Chow–Künneth decomposition modulo algebraic equivalence, so the negative answer to the original question is a rational-equivalence effect."],"supporting_citations":[{"why":"Supplies the nontrivial 'interesting 0-cycle' on $C\\times C$ whose nonzero Chow class is the obstruction transferred to $Y$.","marker":"[13]"},{"why":"Classifies prime Fano threefolds of genus 7 as transverse linear sections of the spinor tenfold, giving the model used throughout.","marker":"[41]"},{"why":"Provides the biregular classification of Fano threefolds that underlies the identification of the genus-7 family.","marker":"[42]"},{"why":"Establishes the homological projective duality and derived-category relation between $Y$ and its dual curve $C$, used for the relative correspondence.","marker":"[20]"},{"why":"Shows the spinor tenfold has trivial Chow groups and one-dimensional $A^2$, yielding the Franchetta property for $Y$.","marker":"[23]"},{"why":"Supplies the stratified projective bundle framework used to compute the generically defined cycles on $Y\\times Y$.","marker":"[8]"},{"why":"Provides the Bloch–Srinivas decomposition-of-diagonal argument used for the motive isomorphism and for the modulo-algebraic-equivalence result.","marker":"[5]"},{"why":"Introduced the notion of multiplicative Chow–Künneth decomposition that the paper rules out for genus-7 Fano threefolds.","marker":"[51]"},{"why":"Supplies structural properties of multiplicative Chow–Künneth decompositions used to force the form of the decomposition in the corollary.","marker":"[10]"}],"fun_headline_variants":["Genus-7 Fano threefold lacks multiplicative Chow-Kunneth splitting","Explicit 2-cycle blocks Chow-Kunneth on genus-7 Fano threefolds","Non-torsion Chow cycle rules out multiplicative decomposition on genus-7","Multiplicative decomposition fails for genus-7 Fano threefold","Fano genus-7: homologically trivial but Chow-nontrivial 2-cycle"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire obstruction transfers through the Chow-level isomorphism of motives between $Y$ and its dual curve $C$; that isomorphism is only sketched, and if the correspondence realizing it fails to be generically defined or to induce the claimed injection on Chow groups, the conclusion collapses.","fun_headline_variants_meta":{"raw":{"variants":["Genus-7 Fano threefold lacks multiplicative Chow-Kunneth splitting","Explicit 2-cycle blocks Chow-Kunneth on genus-7 Fano threefolds","Non-torsion Chow cycle rules out multiplicative decomposition on genus-7","Multiplicative decomposition fails for genus-7 Fano threefold","Fano genus-7: homologically trivial but Chow-nontrivial 2-cycle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001059,"raw_usage":{"total_tokens":4412,"prompt_tokens":881,"completion_tokens":3531,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":3427}},"tokens_in":497,"tokens_out":3531,"duration_ms":23324,"temperature":1.0,"reasoning_tokens":3427,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:53:09.975273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute an explicit rational equivalence in $A^4(Y\\times Y)$ for $Z_Y$ on a single very general genus-7 Fano threefold, or prove that the universal family $Y\\times_B Y$ satisfies the Franchetta property in codimension 4; either would directly contradict Theorem 3.1.","supporting_citations":[{"cited_title":"Green and P","cited_arxiv_id":null,"evidence_quote":"Supplies the nontrivial 'interesting 0-cycle' on $C\\times C$ whose nonzero Chow class is the obstruction transferred to $Y$."},{"cited_title":"Mukai, Curves and Symmetric Spaces, I, American Journal of Mathematics 117, no","cited_arxiv_id":null,"evidence_quote":"Classifies prime Fano threefolds of genus 7 as transverse linear sections of the spinor tenfold, giving the model used throughout."},{"cited_title":"Mukai, Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc","cited_arxiv_id":null,"evidence_quote":"Provides the biregular classification of Fano threefolds that underlies the identification of the genus-7 family."},{"cited_title":"Kuznetsov, Derived categories of Fano threefoldsV 12, Math","cited_arxiv_id":null,"evidence_quote":"Establishes the homological projective duality and derived-category relation between $Y$ and its dual curve $C$, used for the relative correspondence."},{"cited_title":"Kuznetsov, On linear sections of the spinor tenfold, I, Izvestiya: Mathematics 82 no","cited_arxiv_id":null,"evidence_quote":"Shows the spinor tenfold has trivial Chow groups and one-dimensional $A^2$, yielding the Franchetta property for $Y$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the stratified projective bundle framework used to compute the generically defined cycles on $Y\\times Y$."},{"cited_title":"Bloch and V","cited_arxiv_id":null,"evidence_quote":"Provides the Bloch–Srinivas decomposition-of-diagonal argument used for the motive isomorphism and for the modulo-algebraic-equivalence result."},{"cited_title":"Shen and Ch","cited_arxiv_id":null,"evidence_quote":"Introduced the notion of multiplicative Chow–Künneth decomposition that the paper rules out for genus-7 Fano threefolds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies structural properties of multiplicative Chow–Künneth decompositions used to force the form of the decomposition in the corollary."}],"review_version":1}