{"id":"0bff72ea-b6f3-42bc-9809-b70e0b12c981","arxiv_id":"2508.19471","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A faithful group action on a smooth complete intersection of three (1,1) divisors in P^3 x P^3 is linearizable exactly when the group does not mix the two projections, i.e., when the equivariant Picard group has rank 2.","lead":"This paper proves a classification result for symmetry groups acting on a family of Fano threefolds: the group action can be made linear, meaning conjugate to a group of projective transformations, if and only if the group does not mix the two rulings of the threefold. The proof uses the intermediate Jacobian, a birational invariant that detects when a threefold is rational, in an equivariant setting.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of the non-G-Fano direction assumes that every equivariant factorization of X→P^3 contains a blowup of a curve isomorphic to C; this fails when J(C) is non-simple and is also insufficient to force α1=α2.","rationale":"The reader's conditional verdict is appropriate. The key algebraic computation (Proposition 3.4) appears sound, but the final geometric reduction is not. My main concern coincides with the reader's: the inference from Picard rank two and IJ_X=J(C) to a single exceptional curve isomorphic to C is unjustified once J(C) is non-simple. I add that even in the simple case the identification of the G-action on that curve with the natural action is not proved. The formal factorization with two genus-2 blowups and one elliptic blowdown shows the numerical/rank data do not force a genus-3 exceptional curve; hence the step is a genuine gap, not a mere ellipsis. A rigorous proof needs either a simplicity theorem for J(C) in this family or a different argument controlling all possible equivariant factorizations. I therefore keep the verdict CONDITIONAL, as the result may be true and the computation may be correct, but the paper is not complete as written.","tokens_in":8122,"tokens_out":18193,"duration_ms":183133,"concrete_test":"Verify the disputed inference by formalizing the last paragraph with J(C) non-simple. Exhibit a net-rank-one factorization of a rational threefold whose intermediate Jacobian is J(C) but whose exceptional curves have strictly smaller Jacobians: blow up P^3 along two genus-2 curves B1 and B2 and blow down an elliptic curve E, with J(B1)=A1×A2, J(B2)=A3×E; the final IJ is A1×A2×A3, isogenous to a non-simple genus-3 Jacobian, yet no exceptional curve has genus 3. If such a configuration is compatible with the stated hypotheses, the Torelli/decomposition step cannot select a curve isomorphic to C; checking this configuration (e.g., with the Fermat quartic as C) would settle whether the proof step holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The nontrivial direction of Theorem 1.1 rests on the final-paragraph assertion that, because X has Picard rank two and IJ_X≅J(C), Torelli plus uniqueness of the decomposition of ppavs into simple components forces one of the φ_i in an equivariant factorization to be a blowup along a G-invariant curve isomorphic to C. This does not follow. The factorization changes Picard rank by ±1 at each step, so a net change of +1 permits several blowups and blowdowns. The intermediate Jacobian of the final variety is then an alternating sum (product/quotient) of the Jacobians of the contracted curves, not a single summand. Uniqueness of simple components only identifies the isogeny factors of this alternating sum with those of J(C); unless J(C) is simple, no individual curve need have Jacobian isogenous (let alone isomorphic) to J(C). Plane quartics with non-simple Jacobians exist (e.g., the Fermat quartic has Jacobian isogenous to E^3), and the paper neither excludes them nor proves a simplicity statement. Even if such a curve B≅C were found, the transported G-action α1 on C would not automatically coincide with the natural action α2 of §3.1; the final sentence 'α1 and α2 must be the same group action' is therefore also unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies smooth complete intersections X of three divisors of bidegree (1,1) in P^3 × P^3, i.e. Fano threefolds in deformation family №2.12. The main theorem (Theorem 1.1) claims that a faithful action of a group G on X is linearisable if and only if X is not G-Fano, equivalently rk(Pic^G(X)) ≠ 1. The easy direction uses the projection to P^3 when no automorphism swaps the two hyperplane classes. The hard direction assumes X is G-Fano, reduces to the cyclic group generated by an element σ swapping the two classes, and constructs an induced G-action on the plane quartic C over which X is the blow-up of P^3. The paper computes the characters of G on H^0(C,K_C) (Proposition 3.3) and on H^2(X,Ω^1_X) (Proposition 3.4), obtaining representations that differ by a sign. It then seeks a contradiction by assuming a G-equivariant birational map X ⇢ P^3 and using functorial factorization and Torelli's theorem to force an equivariant identification with C.","tokens_in":8387,"tokens_out":42694,"duration_ms":417659,"significance":"If the proof were completed, the result would answer a question of Cheltsov–Li–Ma'u–Pinardin and give a clean linearizability criterion for this family of Fano threefolds, extending the equivariant Clemens–Griffiths technique. The paper contains substantial and mostly careful computations: the geometric description of the family (Proposition 2.1), the cohomological machinery behind Proposition 3.4 (Lemmas 3.5 and 3.7), and the explicit character calculations are valuable and appear reproducible. The main weaknesses are concentrated in the final step of Theorem 1.1, where the Torelli argument is not rigorous, and in a duality/convention issue in the identification of the Lie algebra of the Jacobian. These issues affect the central claim, so the paper needs revision before the theorem can be accepted.","major_comments":[{"comment":"The paper relies on [CLMP24, §1] for the exact sequence describing Aut(X), a preprint by overlapping authors. Please provide either a proof of this fact or a published reference, since it is used in the first paragraph of the proof of Theorem 1.1.","section":"§3, proof of Theorem 1.1, final paragraph"}],"minor_comments":[{"comment":"In the displayed computation of the action on two-forms, the determinant in the denominator is missing from the second and third forms, and 'det(M1+uM2+vM2)' should read 'det(M1+uM2+vM3)'.","section":"§3.1, proof of Proposition 3.3"},{"comment":"The same symbols x,y,z denote the coordinates of the two copies of P^3 in equation (1) and the parameters of the linear system in §3.1. This makes the formulas in Proposition 3.3 hard to follow; consider using λ1,λ2,λ3 for the parameters.","section":"§1.2 and §3.1"},{"comment":"The statement that σ acts on det(xM1+yM2+zM3) with eigenvalue ϖ^{Σ r_i} is correct, but it would be helpful to derive it directly from equation (4); currently the reader must fill in a short matrix computation.","section":"§3.1, proof of Proposition 3.3"},{"comment":"The 'sign' representation is defined by σ ↦ −1; this is a homomorphism only when the order of σ is even. The paper should note explicitly that any lift of the nontrivial element of μ2 in the exact sequence for Aut(X) has even order.","section":"§3.2, Proposition 3.4"},{"comment":"The proof of Proposition 2.1 contains a minor local computation with signs ('y2 = a0 y0 − a1 y1') that is easy to misread; a brief clarification of the sign convention would improve readability.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is of interest, but the final Torelli step is a genuine gap. The Lie algebra convention issue also needs correction because it affects the central character comparison. If the authors can supply the missing equivariant Torelli lemma and fix the duality convention, the paper would be a solid contribution; in its present form I cannot recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the main computation is good, but the last step of Theorem 1.1 has a real gap. The sign calculation in Proposition 3.4 is the genuinely new part and I believe it is correct. The paper answers an explicit open question and the writing is mostly clear.\n\nThe problem is the justification that any equivariant factorization X → P3 must pass through a blow-up of a curve isomorphic to C. The paper argues from Picard rank two, Torelli, and uniqueness of simple components. That does not work as written. A factorization with total Picard change +1 may contain several blow-ups and blow-downs. The intermediate Jacobian of the final model is then an alternating sum of the Jacobians of the contracted curves. If J(C) is not simple, there is no reason any single curve in that sum has Jacobian isomorphic to J(C). The Fermat quartic is an example where J(C) is isogenous to E^3, and the paper neither excludes such curves nor proves a simplicity statement. So the claim that one φ_i is a blow-up of a curve isomorphic to C is unsupported.\n\nThere is a second, smaller issue in the same paragraph. Even if such a curve is found, the G-action α1 transported via the factorization is not automatically the same as the natural action α2 induced on C by the embedding in P3×P3. The sentence 'α1 and α2 must be the same group action' does not follow from the fact that the action on a plane quartic is determined by the action on its canonical space; that fact only goes in one direction. So the contradiction at the end of the proof is not yet established.\n\nIf the authors can justify the factorization step—say, by showing that the total character forces exactly one factor, or by decomposing the representation and using the G-equivariant isogeny structure—the theorem would be sound. As written, the proof is incomplete. The computational core (Propositions 3.3 and 3.4) is careful and I see no issue there. The background from [CLMP24] is appropriate and not circular.\n\nThis paper deserves a serious referee. The result is likely true, and the main computation is a real contribution. But the current version needs a substantial argument added in the final paragraph, and a few typos corrected.\n\nWho should read it: people working on equivariant birational geometry and Cremona groups. I would not cite it until the gap is fixed.","headline":"Nice computation of the equivariant intermediate Jacobian, but the final step of the main theorem has a real gap that needs closing.","tokens_in":8913,"tokens_out":6771,"would_cite":false,"duration_ms":62147,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E08","14J45","14L30","14K30","14H40"],"pacs":[],"model":"deepseek-v4-flash","headline":"A faithful group action on X is linearisable if and only if X is not G-Fano.","keywords":["Fano threefolds","family 2.12","equivariant rationality","linearisable group actions","intermediate Jacobian","plane quartic","Clemens-Griffiths criterion","G-Fano"],"falsifier":"The statement would be refuted by exhibiting a smooth $X$ in family 2.12 and a faithful $G$-Fano action for which an explicit $G$-equivariant birational map $X\\dashrightarrow\\mathbb{P}^3$ exists. Concretely, for the involution $\\sigma$ that swaps the two $\\mathbb{P}^3$ factors, write the defining equations with $\\sigma$-invariant matrices, compute the eigenvalues of $\\sigma$ on $H^0(C,K_C)$ and on $H^2(X,\\Omega^1_X)$; Proposition 3.4 predicts they differ by the factor $-1$. If an explicit computation on any example produced equal multisets, the non-isomorphism claim would fail, and one should then search for an equivariant map by eliminating variables.","tokens_in":7904,"feed_emoji":"🔀","tokens_out":10859,"duration_ms":111839,"temperature":0.7,"pith_summary":"This paper proves a complete dichotomy for faithful group actions on the Fano threefolds in deformation family 2.12, the smooth complete intersections of three divisors of bidegree $(1,1)$ in $\\mathbb{P}^3\\times\\mathbb{P}^3$. For any such threefold $X$ and any faithfully acting group $G$, the action is linearisable — conjugate to a linear action on $\\mathbb{P}^3$ — exactly when $X$ is not $G$-Fano, i.e. when $\\mathrm{rk}(\\mathrm{Pic}^G(X))\\neq 1$. The easy half is the blow-down $X\\to\\mathbb{P}^3$ along the genus-three curve $C$, which is $G$-equivariant precisely outside the $G$-Fano case. The hard half shows that in the $G$-Fano case a $G$-equivariant birational map to $\\mathbb{P}^3$ would force an equivariant isomorphism between the intermediate Jacobian of $X$ and the Jacobian of $C$, while a direct character computation makes those two representations differ by the sign character. This gives a genuinely equivariant use of the Clemens\\textendash Griffiths intermediate Jacobian obstruction.","feed_headline":"Fano threefold symmetries linearise exactly when not G-Fano","feed_subtitle":"A sign mismatch between the intermediate Jacobian and the curve's Jacobian forces the non-linear case.","key_machinery":"The central mechanism is the comparison of two three-dimensional $G$-representations: $\\mathrm{Lie}(IJ_X)=H^2(X,\\Omega^1_X)$ and $\\mathrm{Lie}(J_C)=H^0(C,K_C)$. Proposition 3.4 computes the first via the conormal sheaf sequence and a Koszul resolution, obtaining a character sum tensored with the sign representation; Proposition 3.3 computes the second directly from the plane quartic model. Together with the Clemens\\textendash Griffiths and Torelli identifications identifying $IJ_X$ with $J_C$ for blow-ups of $\\mathbb{P}^3$ along curves, this sign twist is the obstruction that kills linearisability in the $G$-Fano case.","core_discovery":"Let $X$ be a smooth complete intersection of three divisors of bidegree $(1,1)$ in $\\mathbb{P}^3\\times\\mathbb{P}^3$, and let $G$ act faithfully on $X$. Theorem 1.1 characterises linearisability by the rank of the invariant Picard group: the $G$-action is linearisable if and only if $\\mathrm{rk}(\\mathrm{Pic}^G(X))\\neq 1$. Since the blow-up structure $X\\to\\mathbb{P}^3$ gives $\\mathrm{Pic}(X)=\\mathbb{Z}H\\oplus\\mathbb{Z}E$, the $G$-Fano condition $\\mathrm{Pic}^G(X)=\\mathbb{Z}[-K_X]$ is equivalent to the existence of an element $\\sigma\\in G$ that swaps the two hyperplane classes $H$ and $H'$. The core of the proof is a sign computation: writing $C$ for the genus-three plane quartic over which $X$ is the blow-up of $\\mathbb{P}^3$, the induced $G$-representation on $\\mathrm{Lie}(J_C)=H^0(C,K_C)$ is $\\bigoplus_j\\chi^{s_j+\\sum s_i-\\sum r_i}$, while the action on the intermediate Jacobian $\\mathrm{Lie}(IJ_X)=H^2(X,\\Omega^1_X)$ is the same representation tensored with the sign character (Proposition 3.4). A $G$-equivariant birational map to $\\mathbb{P}^3$ would have to factor through a blow-up along a $G$-invariant curve isomorphic to $C$, forcing $\\mathrm{Lie}(IJ_X)$ to be equivariantly isomorphic to $\\mathrm{Lie}(J_C)$; the two representations differ by a sign, so no such map exists.","pith_inferences":["The same sign-twist mechanism should obstruct linearisability for any Fano threefold that is a blow-up of $\\mathbb{P}^3$ along a curve whose intermediate Jacobian is the curve's Jacobian, with the sign arising whenever the ambient presentation has two symmetric rulings.","A natural testable extension is to compute the same character for the second projection curve $C'$; since $C'$ is isomorphic to $C$, the same twisted representation should appear, showing that the obstruction is independent of the chosen blow-down.","If the Jacobian of $C$ decomposes into simple factors, the Torelli-based identification of the blow-up centre would need a more refined treatment, but the representation-theoretic obstruction given by Proposition 3.4 is independent of that decomposition and would still have to be matched by any successful linearisation."],"forward_implications":["For every faithful action on $X$, the equivariant birational type is one of two kinds: either the group acts linearly on $\\mathbb{P}^3$, or the invariant Picard group is exactly $\\mathbb{Z}[-K_X]$; there is no intermediate possibility.","The theorem answers the question recorded in [CLMP24, Remark 6] affirmatively.","Any element of $\\mathrm{Aut}(X)$ that swaps the two $\\mathbb{P}^3$ factors generates a cyclic non-linearisable action, giving explicit examples of non-linear actions on a rational threefold.","The proof demonstrates that the Clemens\\textendash Griffiths intermediate Jacobian obstruction works in an equivariant setting for geometric group actions over an algebraically closed field, not only for Galois actions over non-closed fields.","In the non-$G$-Fano case, the blow-down $X\\to\\mathbb{P}^3$ is $G$-equivariant, so linearisability is achieved geometrically without changing the model."],"supporting_citations":[{"why":"Poses the question answered by Theorem 1.1 and supplies the exact sequence describing $\\mathrm{Aut}(X)$ with its $\\mu_2$ quotient.","marker":"[CLMP24]"},{"why":"Provides the construction of $X$ from a plane quartic curve, the blow-up description, and the Hilbert polynomial computation identifying $C$ as a genus-three curve.","marker":"[Dol12]"},{"why":"Introduces the intermediate Jacobian obstruction to rationality that the paper uses in equivariant form.","marker":"[CG72]"},{"why":"Supplies the lemma identifying the intermediate Jacobian of a blow-up of $\\mathbb{P}^3$ along a curve with the curve's Jacobian.","marker":"[PS99]"},{"why":"Gives the functorial factorization of birational maps into equivariant blow-ups and blow-downs used to decompose a hypothetical $G$-linearisation.","marker":"[AT19]"},{"why":"Provides the alternate equivariant factorization statement invoked alongside [AT19].","marker":"[KT22]"},{"why":"Contains the character computation for Jacobians of plane quartics that Proposition 3.3 follows.","marker":"[CTZ25]"},{"why":"Supplies the hypercohomology spectral sequence and K\\\"unneth formula used to compute $H^3(L_i)$ as $G$-representations.","marker":"[Sta24]"}],"fun_headline_variants":["Only G-Fano actions block linearisation","When Picard rank is one, linearisation fails","Non-G-Fano? Then linearise it","Sign mismatch kills equivariant linearisation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the claim that every equivariant birational map from $X$ to $\\mathbb{P}^3$ must factor through a blow-up of a $G$-invariant curve whose Jacobian is the same simple Jacobian as $C$; if the Jacobian of $C$ splits into smaller pieces, that step is not automatic and the proof would need a separate argument.","fun_headline_variants_meta":{"raw":{"variants":["Only G-Fano actions block linearisation","When Picard rank is one, linearisation fails","Non-G-Fano? Then linearise it","Sign mismatch kills equivariant linearisation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001729,"raw_usage":{"total_tokens":6836,"prompt_tokens":945,"completion_tokens":5891,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":5834}},"tokens_in":561,"tokens_out":5891,"duration_ms":41526,"temperature":1.0,"reasoning_tokens":5834,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:54:37.637815+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The statement would be refuted by exhibiting a smooth $X$ in family 2.12 and a faithful $G$-Fano action for which an explicit $G$-equivariant birational map $X\\dashrightarrow\\mathbb{P}^3$ exists. Concretely, for the involution $\\sigma$ that swaps the two $\\mathbb{P}^3$ factors, write the defining equations with $\\sigma$-invariant matrices, compute the eigenvalues of $\\sigma$ on $H^0(C,K_C)$ and on $H^2(X,\\Omega^1_X)$; Proposition 3.4 predicts they differ by the factor $-1$. If an explicit computation on any example produced equal multisets, the non-isomorphism claim would fail, and one should then search for an equivariant map by eliminating variables.","supporting_citations":[],"review_version":1}