{"id":"75382926-1e12-4034-8843-28c8ab6de7b6","arxiv_id":"2412.02256","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For K3 surfaces, every primitive base point free g^r_d with d≥4, r≥sqrt(d/2), and g>2d-3+(r-1)^2 has a Donagi-Morrison lift N adapted to |C| with Cliff(N⊗O_C)≤Cliff(A).","lead":"The paper proves a new sufficient condition under which a special line bundle on a curve lying on a K3 surface can be lifted to a line bundle on the surface itself. This makes progress on a 1989 Donagi-Morrison conjecture about when such lifts exist, without requiring the surface polarization to be ample.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's key structural step, Lemma 4.4, is imported from [14] without statement or proof; if its hypotheses are not met in the non-ample setting of Theorem 1.2, the main theorem is unsupported.","rationale":"The reader's verdict and weakest-assumption analysis align with my stress-test. The single most load-bearing point is the unproved Lemma 4.4. The rest of the proof is internally consistent: the numerical hypothesis g > 2d - 3 + (r-1)^2 is used to force rho < 0 and to make the inequalities in Lemma 4.1 and Proposition 4.2 work; the Chern class computation (5) is correct under the standard convention for torsion-free sheaves; the adaptation argument and the Clifford inequality are valid conditional on Lemma 4.4 and the auxiliary results. I examined the N^2 = 0 and N^2 > 0 cases and found no circularity or fitted-parameter reasoning. The only way the central claim could fail is if Lemma 4.4 does not hold in the intended non-ample setting, or requires hypotheses not present in Theorem 1.2. Since the lemma is imported from [14] and not stated, this cannot be checked from the paper alone. Therefore a conditional acceptance—pending verification of [14, Lemma 4.1]'s hypotheses and proof in this setting—is the appropriate verdict. I recommend no change to the reader's verdict.","tokens_in":12479,"tokens_out":17782,"duration_ms":163811,"concrete_test":"Obtain [14] and check whether the hypotheses of its Lemma 4.1 are satisfied under the assumptions of Theorem 1.2: X a K3, L base point free and big (not assumed ample), C ∈ |L| smooth, A primitive base point free g^r_d with d >= 4, r >= sqrt(d/2), and g > 2d - 3 + (r-1)^2. Verify that the proof of [14, Lemma 4.1] constructs a saturated line subbundle M with h0(M) >= 2 and a globally generated torsion-free quotient F without using ampleness of L (e.g., without requiring |L| to have no fixed components). If [14, Lemma 4.1] has extra hypotheses, or if its proof uses L ample in an essential way, then Lemma 4.4 is unsupported and the theorem needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2 hinges on Lemma 4.4, which asserts that the Lazarsfeld-Mukai bundle E_{C,A} admits a saturated line subbundle M with h0(M) >= 2 such that the quotient F has rank r, is torsion-free, and is globally generated (exact sequence (3)). This decomposition is used at every subsequent step: equation (5) computes d from M, ell(W), and c2(F^∨∨); the definition N = c1(F^∨∨) and its base-point-freeness rely on F being globally generated; adaptedness of N uses the exact sequence 0 -> M^∨ -> N -> N⊗O_{C'} -> 0; and the Clifford inequality Cliff(N⊗O_C) <= Cliff(A) follows from c2(F^∨∨) >= 2(r-1). The paper does not prove Lemma 4.4; it says only 'by the same way of the proof of Lemma 4.1 as in [14]', without stating that lemma or its hypotheses. The lemma is not a triviality: while a quotient of a globally generated sheaf is globally generated, the nontrivial part is the existence of a saturated line subbundle with h0 >= 2 whose quotient remains globally generated. If [14, Lemma 4.1] requires L ample, or any additional condition beyond the hypotheses of Theorem 1.2 (L base point free and big, A primitive, g > 2d - 3 + (r-1)^2, r >= sqrt(d/2)), then Lemma 4.4 is not justified in the non-ample setting and the proof of the main theorem collapses. No internal inconsistency is apparent in the rest of the argument; the concern is specifically the missing verification of this imported structural result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a quantitative sufficient condition for the existence of Donagi-Morrison lifts of line bundles on curves lying on K3 surfaces, without assuming the polarization L is ample. The main theorem (Theorem 1.2) states that if X is a K3 surface, L is base point free and big, C is a smooth curve in |L| of genus g, and A is a base point free primitive g^r_d on C with d ≥ 4, r ≥ sqrt(d/2), and g > 2d - 3 + (r-1)^2, then there is a line bundle N on X, adapted to |L|, such that |A| ⊂ |N⊗O_C| and Cliff(N⊗O_C) ≤ Cliff(A). The proof uses the Lazarsfeld-Mukai bundle E_{C,A}, a decomposition into a line subbundle and a torsion-free quotient (Lemma 4.4), Chern class identities, and case analysis for the N^2 = 0 elliptic-fiber case. A corollary is drawn for line bundles computing the special Clifford index.","tokens_in":12782,"tokens_out":16870,"duration_ms":154557,"significance":"If correct, the result is a genuine advance: it extends the Donagi-Morrison lift theorem to polarizations that are only base point free and big, under an explicit numerical inequality, and it provides a new sufficient condition that is checkable. The proof strategy is elegant, particularly the use of the Chern class identity c2(E) = M.N + ℓ(W) + c2(F∨∨) together with nonnegativity of Cliff(F∨∨) to obtain the Clifford inequality. The paper is short and mostly self-contained, with clearly stated hypotheses and a concrete corollary for the special Clifford index. However, the central structural lemma (Lemma 4.4) is imported from the author's previous work [14] without proof or statement, and at least one numerical argument in Lemma 4.1 is under-justified; these gaps must be addressed before the theorem can be considered fully established.","major_comments":[{"comment":"Lemma 4.4 is the central structural input: it asserts that E_{C,A} has a saturated line subbundle M with h0(M) ≥ 2 whose quotient F is torsion free of rank r and globally generated. The proof of Theorem 1.2 depends on this decomposition at every stage: equation (3), the definition N = c1(F∨∨), the base-point-freeness and adaptedness of N, and the final bound c2(F∨∨) ≥ 2(r−1) all rely on it. However, the lemma is neither stated nor proved in the present paper; the text says only that it follows 'by the same way of the proof of Lemma 4.1 as in [14]'. The statement and hypotheses of [14, Lemma 4.1] are not reproduced, and there is no verification that its assumptions are satisfied in the setting of Theorem 1.2, where L is only base point free and big (not necessarily ample). Please provide a complete proof of Lemma 4.4, or state [14, Lemma 4.1] explicitly and check its hypotheses in detail for the present non-ample situation.","section":"Section 4, Lemma 4.4"},{"comment":"In the proof of Lemma 4.1, after the displayed lower bounds for (C−r∆−kΓ1)^2 and C·(C−r∆−kΓ1), the conclusion h0(L⊗O_X(−r∆−kΓ1)) > 2 is stated to follow from the Riemann-Roch theorem because C is nef. This step is not immediate: the divisor H := C − r∆ − kΓ1 is not shown to be nef, and in fact H·Γ1 ≤ −2 can occur (for k = 0 when C·Γ1 ≤ r−2), so h1(H) need not vanish. The Riemann-Roch formula alone gives h0(H) − h1(H) + h0(H∨) = 2 + H^2/2, which does not yield h0(H) > 2 unless h1(H) is controlled. Since Lemma 4.1 feeds directly into Proposition 4.2 and hence into the N^2 = 0 case of Theorem 1.2, this gap is load-bearing. Please supply a missing argument (for example, a suitable vanishing theorem or a subtraction of (−2)-curves) to justify the claimed lower bound on h0.","section":"Section 4, Lemma 4.1"}],"minor_comments":[{"comment":"There is a typo: 's ee [1], [2]' should read 'see [1], [2]'.","section":"Introduction, p.3"},{"comment":"The deduction 'by Proposition 2.4, we have D^2 = 0' is not explained; Proposition 2.4 states 2-connectedness of members of |L| and does not directly imply D^2 = 0. Please add a short justification (e.g., using the hyperbolic lattice structure of NS(X) and the fact that D is a movable, hence nef, divisor orthogonal to an isotropic class).","section":"Section 4, proof of Lemma 4.2"},{"comment":"Equation (5) is quoted from [7, (0.3)]. For self-containedness, please derive it from the exact sequences (3) and (4) together with the Riemann-Roch theorem and the additivity of Chern classes.","section":"Section 4, proof of Theorem 1.2"},{"comment":"The wording is confusing: condition (i) of Conjecture 1.1 is the containment condition, whereas the adaptedness conditions are defined separately in the introduction. Please rephrase to clarify that when h1(L⊗N∨) = 0, the restriction map H0(N) → H0(N⊗O_C) is surjective, so that |N⊗O_C| is the restriction of |N|.","section":"Remark 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper relies crucially on [14], which is by the author and a collaborator. It would be prudent to ask the author to verify explicitly that [14, Lemma 4.1] applies verbatim to the base-point-free-and-big case, or to include its full statement and proof. The gap in Lemma 4.1 concerning the derivation of h0 > 2 should also be addressed. The paper's scope is appropriate for a 'remark' and the main idea is sound, so I expect the issues to be fixable within the manuscript's framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I'll keep this short. The paper proves a genuine extension of the Donagi-Morrison conjecture: when L is base point free and big (not assumed ample), and g > 2d - 3 + (r-1)^2, any primitive g^r_d with d≥4 and r≥sqrt(d/2) has a Donagi-Morrison lift. Previous results either assumed L ample or handled only nets. The numerical condition is a hypothesis, not an output, so there is no circularity in that sense.\n\nThe proof strategy is sound. The Lazarsfeld-Mukai bundle is decomposed, Chern classes are computed, and the two cases N^2=0 and N^2>0 are handled separately. The Clifford inequality at the end follows from c2(F^∨∨)≥2(r-1). The case analysis for the elliptic-curve situation (Lemmas 4.1–4.3) is worked out in detail and looks coherent.\n\nThe soft spot is Lemma 4.4. It asserts that E_{C,A} has a saturated line subbundle M with h0(M)≥2 whose quotient F is rank r, torsion free, and globally generated. Everything downstream depends on this: the definition N=c1(F^∨∨), the adaptedness argument, and the Clifford bound. The proof says only 'by the same way of the proof of Lemma 4.1 as in [14]', with no statement of that lemma or its hypotheses. If [14] required L ample, the theorem as stated is unsupported. This is a real gap, and the paper signals it by omitting the proof. I can't tell from the text whether the lemma carries over to the non-ample setting; the author likely knows, but a referee needs to see it.\n\nMinor point: Example 4.1 is imported from [14] and not new, which is fine.\n\nNet: the central idea is good and the main theorem is probably true, but the paper is not self-contained at its critical step. That is a fixable problem—state Lemma 4.4 with full hypotheses and give a proof or a precise citation—rather than a fatal flaw. I'd send this to a serious referee, asking specifically whether Lemma 4.4 holds under the hypotheses of Theorem 1.2. The rest of the argument deserves scrutiny but passes a first read.","headline":"First Donagi-Morrison lift result for non-ample big line bundles, with a proof that hinges on an unproved imported lemma.","tokens_in":13417,"tokens_out":2290,"would_cite":true,"duration_ms":22152,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J28","14J60","14H60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For base point free primitive line bundles on K3 curves satisfying $g>2d-3+(r-1)^2$, a Donagi-Morrison lift always exists.","keywords":["K3 surfaces","Donagi-Morrison conjecture","Lazarsfeld-Mukai bundles","Clifford index","Brill-Noether theory","line bundles on curves","adapted line bundles","special Clifford index"],"falsifier":"Compute $E_{C,A}$ for the double cover of a smooth plane quartic described in Example 4.1 (degree 8, rank 2, genus satisfying the inequality) and check whether it has a saturated line subbundle $M$ with $h^0(M)\\ge2$ and a torsion free globally generated quotient $F$ of rank 2. If any such example lacks the decomposition, Lemma 4.4 and hence the proof of Theorem 1.2 fail; if it holds in all small cases, the key step is supported.","tokens_in":12174,"feed_emoji":"📐","tokens_out":12672,"duration_ms":111646,"temperature":0.7,"pith_summary":"This paper proves a quantitative existence result for a conjecture about curves on K3 surfaces: a special line bundle on a curve should be the restriction of a line bundle on the surface. The theorem says that if the line bundle is base point free, primitive, of degree $d\\ge 4$ and rank $r\\ge \\sqrt{d/2}$, and the curve's genus satisfies $g>2d-3+(r-1)^2$, then such a surface line bundle $N$ exists, is adapted to $|C|$, contains the original linear system, and has no larger Clifford index. The significance is that $N$ is not required to be ample, so the result covers big linear systems on arbitrary K3 surfaces. The result therefore provides a numerical threshold for Donagi-Morrison lifts and a step toward the full conjecture.","feed_headline":"K3 special line bundles get surface lifts past a genus bound","feed_subtitle":"For g > 2d−3+(r−1)^2, every base-point-free primitive g^r_d on a K3 curve has a Donagi-Morrison lift.","key_machinery":"The central object is the Lazarsfeld-Mukai bundle $E_{C,A}$, a rank $r+1$ vector bundle on $X$ with $c_1(E_{C,A})=\\mathcal{O}_X(C)$ and $c_2(E_{C,A})=d$, obtained from the evaluation map $H^0(A)\\otimes\\mathcal{O}_X\\to A$. The proof's load-bearing step is a decomposition $0\\to M\\to E_{C,A}\\to F\\to0$ with $M$ a saturated line subbundle, $h^0(M)\\ge2$, and $F$ a torsion free globally generated rank-$r$ sheaf; the lift is $N=c_1(F^{\\vee\\vee})$. Adaptedness to $|L|$ is established through exact sequences and vanishing, and the Clifford comparison follows from $d=M.N+\\ell(W)+c_2(F^{\\vee\\vee})$, the bound $\\operatorname{Cliff}(F^{\\vee\\vee})\\ge0$, and $h^0(N\\otimes\\mathcal{O}_C)\\ge h^0(N)$.","core_discovery":"The paper establishes Theorem 1.2: for a K3 surface $X$, a base point free big line bundle $L$, a smooth curve $C\\in|L|$ of genus $g$, and a base point free primitive line bundle $A=g^r_d$ with $d\\ge 4$ and $r\\ge \\sqrt{d/2}$, the inequality $g>2d-3+(r-1)^2$ implies that $|A|$ is contained in $|N\\otimes\\mathcal{O}_C|$ for some line bundle $N$ on $X$ adapted to $|L|$, with $\\operatorname{Cliff}(N\\otimes\\mathcal{O}_C)\\le\\operatorname{Cliff}(A)$. This is a Donagi-Morrison lift in the case where $L$ is big but not necessarily ample, and Corollary 4.1 extends the conclusion to line bundles computing the special Clifford index of $C$.","pith_inferences":["The threshold $g>2d-3+(r-1)^2$ is an artifact of the proof's estimates and Hodge-index contradictions; it is likely not sharp, so the true range of Donagi-Morrison lifts may be substantially wider.","Because Lemma 4.4 is imported without proof, a natural test is to prove the same saturation decomposition for all primitive line bundles with negative Brill-Noether number; if true, the theorem would extend to the entire negative Brill-Noether range.","The adaptedness argument suggests a general mechanism: whenever $h^1(M^\\vee)$ is constant as $C'$ varies in $|L|$, the surface line bundle $N=c_1(F^{\\vee\\vee})$ is adapted; this could yield a standalone criterion for adaptedness."],"forward_implications":["For every primitive base point free $g^r_d$ with $d\\ge4$, $r\\ge\\sqrt{d/2}$, and $g>2d-3+(r-1)^2$, there is a line bundle $N$ on $X$ adapted to $|L|$ with $|A|\\subset|N\\otimes\\mathcal{O}_C|$ and $\\operatorname{Cliff}(N\\otimes\\mathcal{O}_C)\\le\\operatorname{Cliff}(A)$.","This is the first such lift theorem that does not require $L$ to be ample, so it applies to big linear systems on K3 surfaces of arbitrary Picard number.","Corollary 4.1 gives the same conclusion for line bundles computing the special Clifford index of $C$, since those are primitive and have negative Brill-Noether number.","In the $N^2=0$ case, the proof forces $A\\cong\\mathcal{O}_C(r\\Delta)$ for an elliptic curve $\\Delta$, so the elliptic case is completely settled inside the theorem."],"supporting_citations":[{"why":"States the Donagi-Morrison conjecture and the three conditions that a lift must satisfy, which the theorem addresses.","marker":"[3]"},{"why":"Supplies the theory of generalized Lazarsfeld-Mukai bundles, including the Clifford-index bound $\\operatorname{Cliff}(E)\\ge0$ used in the final inequality.","marker":"[11]"},{"why":"Lemma 4.1 of this reference is imported as Lemma 4.4, the saturation/decomposition step on which the whole proof depends.","marker":"[14]"},{"why":"Gives the Chern-class formula $d=M.N+\\ell(W)+c_2(F^{\\vee\\vee})$ that connects the lift $N$ to the degree of $A$.","marker":"[7]"},{"why":"Provides Saint-Donat's classification and the 2-connectedness results used to prove $N$ is base point free and adapted.","marker":"[13]"},{"why":"Provides the vanishing criterion for big line bundles featuring $(-2)$-divisors, used in deriving the elliptic-curve case.","marker":"[8]"},{"why":"Documents the exceptional failure of independence of $h^0(N\\otimes\\mathcal{O}_{C'})$ that motivates the adaptedness condition and gives earlier net cases.","marker":"[12]"}],"fun_headline_variants":["Donagi-Morrison lift proven for big K3 line bundles","Genus bound forces K3 surface lifts for primitive curve bundles","For large genus, every primitive bundle on a K3 curve lifts","K3 curve bundles lift to surfaces past a quadratic genus bound","Donagi-Morrison holds for big line bundles on K3 surfaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on Lemma 4.4, imported from an earlier paper without proof, which says that the bundle $E_{C,A}$ built from $C$ and $A$ splits into a line subbundle with at least two sections and a rank-$r$ quotient generated by its global sections; if that lemma needs extra hypotheses, the main theorem has no support.","fun_headline_variants_meta":{"raw":{"variants":["Donagi-Morrison lift proven for big K3 line bundles","Genus bound forces K3 surface lifts for primitive curve bundles","For large genus, every primitive bundle on a K3 curve lifts","K3 curve bundles lift to surfaces past a quadratic genus bound","Donagi-Morrison holds for big line bundles on K3 surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000339,"raw_usage":{"total_tokens":1843,"prompt_tokens":887,"completion_tokens":956,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":867}},"tokens_in":503,"tokens_out":956,"duration_ms":9000,"temperature":1.0,"reasoning_tokens":867,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:39:54.743992+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $E_{C,A}$ for the double cover of a smooth plane quartic described in Example 4.1 (degree 8, rank 2, genus satisfying the inequality) and check whether it has a saturated line subbundle $M$ with $h^0(M)\\ge2$ and a torsion free globally generated quotient $F$ of rank 2. If any such example lacks the decomposition, Lemma 4.4 and hence the proof of Theorem 1.2 fail; if it holds in all small cases, the key step is supported.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the Donagi-Morrison conjecture and the three conditions that a lift must satisfy, which the theorem addresses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theory of generalized Lazarsfeld-Mukai bundles, including the Clifford-index bound $\\operatorname{Cliff}(E)\\ge0$ used in the final inequality."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lemma 4.1 of this reference is imported as Lemma 4.4, the saturation/decomposition step on which the whole proof depends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Chern-class formula $d=M.N+\\ell(W)+c_2(F^{\\vee\\vee})$ that connects the lift $N$ to the degree of $A$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Saint-Donat's classification and the 2-connectedness results used to prove $N$ is base point free and adapted."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the vanishing criterion for big line bundles featuring $(-2)$-divisors, used in deriving the elliptic-curve case."},{"cited_title":"H., Pencils and nets on curves arising from rank 1 sheaves on K3 surfaces , Math","cited_arxiv_id":null,"evidence_quote":"Documents the exceptional failure of independence of $h^0(N\\otimes\\mathcal{O}_{C'})$ that motivates the adaptedness condition and gives earlier net cases."}],"review_version":1}