{"id":"5b2d9204-2752-4196-b6f6-94a15ec143e6","arxiv_id":"2511.10327","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims a non-isotrivial pencil of plane quartics with one base point, irreducible members, and smooth general member, via 16-torsion on an elliptic normal quartic.","lead":"This paper develops a theory of divisors cut on space curves by cones and uses it to claim a pencil of plane quartics with a single base point and smooth, irreducible, non-isotrivial members. The advertised construction extends a known cubic-curve example to degree four, but the final proof rests on a false lemma.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7.3 is false and is load-bearing: Theorem 7.5 uses it to force Φ''^{-1}(16q) to meet the open cone locus, and without that the pencil in Theorem 7.8 has no certified cone with vertex in U.","rationale":"We read the paper in good faith. Sections 2–6 develop a plausible framework: the limit computations, the cone map, and the differential analysis give evidence of substantial work. The main theorem, however, is an existence statement about pencils of plane quartics, and the proof passes through Theorem 7.5. That proof's crucial step is the application of Lemma 7.3. The lemma as stated is false. Although the reader's proposed P^1×P^1 example is not a morphism at two points, replacing it with the blow-up construction above gives a genuine counterexample with smooth projective varieties. Consequently the dimension statement 'dim μ(G)≥1' does not follow, and the subsequent argument that μ(G)∩U is non-empty (via excluding μ(G)⊂S∪C using Corollary 6.16) has no basis. Without a cone with vertex in U, the 'fix one such cone' step in Theorem 7.8 cannot be justified, so the existence of the pencil is not established by the paper. This is an internal logical gap, not a disagreement with community consensus. Because the central claim depends on a false lemma, we agree with the reader's REJECT; the framework could be repaired if a replacement proof of the open-cone intersection is found, but as written the main application is unsupported.","tokens_in":25271,"tokens_out":14564,"duration_ms":171139,"concrete_test":"Verify the counterexample: let Y=P^2, X=Bl_p(Y×P^1), φ=pr_1∘σ, μ=pr_{P^1}∘σ (where σ is the blow-down). Check that for y0=pr_1(p), the fiber φ^{-1}(y0) is E∪C with E the exceptional divisor and C the strict transform of {y0}×P^1, and that μ maps both E and C to the single point pr_2(p), while for y≠y0 the fiber maps onto P^1. This directly disproves Lemma 7.3. Then locate the invocation of Lemma 7.3 in the proof of Theorem 7.5 and confirm that no other argument establishes μ(G)∩U≠∅; if none exists, the proof of Theorem 7.8 collapses at that step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised application depends on Theorem 7.5, whose proof invokes Lemma 7.3 to conclude dim μ(G)≥1 and then μ(G)∩U≠∅ for G=(Φ'')^{-1}(16q). Lemma 7.3 is not just unproved; it is false. The proof's step 'a proper subvariety in an affine U has dimension zero' is invalid. A valid counterexample: take Y=P^2, X=Bl_p(Y×P^1), φ:X→Y the first projection, and μ:X→P^1 the composition of the blow-down with the second projection. For general y∈Y the fiber φ^{-1}(y)≅P^1 maps isomorphically onto P^1, so dim μ(φ^{-1}(y))=dim φ^{-1}(y)=1. Over y0=pr_1(p), the fiber is the union of the strict transform of {y0}×P^1 and the exceptional divisor E; both are contracted by μ to the single point pr_2(p), so μ(φ^{-1}(y0)) is a point. Thus the lemma's conclusion fails. (The P^1×P^1 multiplication map sometimes proposed is not regular; the blow-up example avoids that.) Since Lemma 7.3 is the only mechanism in Theorem 7.5 that forces the inverse image of D=16q to contain cones with vertex in the open set U, Theorem 7.5, and the pencil construction in Theorem 7.8 that starts from one such cone, are unsupported as written. The surrounding conic-linear-series framework may be salvageable, but the main existence theorem lacks a proof at this point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of 'conic linear series' on a smooth nondegenerate curve C⊂P^3: linear systems cut out on C by cones of fixed degree with vertex outside C, together with their limits when the vertex specializes to C. Sections 3–6 contain substantial foundational work: an explicit description of linear limits of cones (Proposition 3.4), a projective model of the blow-up of P^3 along C (Proposition 3.16), computations of the limiting conic linear systems R^ℓ_k(p) (Propositions 4.6 and 4.8), the cone map ρ_k and a differential analysis leading to generically maximal-rank statements for the map Φ (Theorem 6.8), and criteria for dominance/surjectivity of Φ and its extension Φ'' (Corollaries 6.9 and 6.14). The advertised application is Section 7: using the elliptic normal quartic C⊂P^3, the authors claim that every effective degree-16 divisor on C is cut out by a 1-dimensional family of quartic cones with vertex in the open locus U (Theorem 7.5), and from this construct a non-isotrivial pencil of plane quartics with one base point, all irreducible and with smooth general member (Theorem 7.8). The main application depends on Lemma 7.3, a general fiber-dimension statement for proper surjective morphisms, which is false.","tokens_in":25660,"tokens_out":5457,"duration_ms":56241,"significance":"If the main theorem were correct, it would provide a new existence result for pencils of plane quartics with strong geometric properties, and the surrounding framework of conic linear series, cone maps, and their differentials could be a useful tool for further study of divisors on curves in P^3. The paper contains several concrete and apparently correct contributions: the explicit limit formula in Proposition 3.4, the description of R^ℓ_{d-1}(p) and R^ℓ_d(p) in Propositions 4.6 and 4.8, the injectivity result for the cone map in Proposition 5.5, and the generically maximal-rank statement for Φ in Theorem 6.8. These parts are developed with detailed, often coordinate-based, arguments. However, the central geometric application rests on Lemma 7.3, and that lemma is not merely unproved but false. Consequently the proof of Theorem 7.5, and with it the construction in Theorem 7.8, is unsupported as written. The failure is load-bearing, not a presentation issue.","major_comments":[{"comment":"Lemma 7.3 is false, and the proof is invalid. The step 'a proper subvariety in the affine U, so it has dimension zero' is wrong: a proper subvariety of an affine variety can have positive dimension, and the image μ(φ^{-1}(y')) need not even be proper. A concrete counterexample is Y=P^2, X=Bl_p(Y×P^1), φ:X→Y the first projection, and μ:X→P^1 the composition of the blow-down with the second projection. For general y∈Y, φ^{-1}(y)≅P^1 and μ maps it isomorphically onto P^1, so dim μ(φ^{-1}(y))=dim φ^{-1}(y)=1. But for y0=pr_1(p), the fiber is the union of the strict transform of {y0}×P^1 and the exceptional divisor; both are contracted by μ to the single point pr_2(p). Thus dim μ(φ^{-1}(y0))=0, contradicting the lemma's conclusion.","section":"Lemma 7.3"},{"comment":"Theorem 7.5 uses Lemma 7.3 as the only mechanism to conclude that μ(G)∩U≠∅ for G=(Φ'')^{-1}(D). The proof applies the false lemma to deduce dim μ(G)≥1, and then argues by contradiction that μ(G) cannot lie in S∪C. Without Lemma 7.3, the argument gives no reason why a component of G should map into U. Since Theorem 7.8 begins by invoking Theorem 7.5 to obtain a 1-dimensional family of quartic cones K_t⊂P_U(E) cutting D=16q on C, the advertised pencil construction is unsupported. A replacement argument is needed that genuinely forces the inverse image of 16q to contain cones with vertex in U.","section":"Theorem 7.5"},{"comment":"Even setting aside the falsity of Lemma 7.3, its hypotheses are not verified in the proof of Theorem 7.5. The sentence 'It is clear that for p∈U the hypothesis in (7.4) holds' confuses the variable p∈U (a vertex of a cone) with the variable y∈Y=P(S_4) (a divisor on C). One would need to prove that for a general divisor D∈P(S_4), dim μ(Φ''^{-1}(D)) = dim Φ''^{-1}(D) (=1). This is a substantive statement about the family of fibers of Φ'', not an immediate consequence of the construction, and no proof is supplied.","section":"Theorem 7.5, hypothesis of Lemma 7.3"}],"minor_comments":[{"comment":"The notation is inconsistent: the proof uses both φ and ϕ for the morphism X→Y, and 'The set ϕ(μ^{-1}(U))' should read 'φ(μ^{-1}(U))'.","section":"Lemma 7.3 proof"},{"comment":"Shortly after Proposition 3.16, the text introduces 'd eΣ−E' and then refers to 'eH'; this seems to be a typo for the pullback of a hyperplane divisor, denoted eL elsewhere in the proof.","section":"Section 3.2"},{"comment":"The phrase 'there is at least a 1-dimensional complete subvariety B of G' is slightly imprecise: the authors later intersect with P_U(E), but the notation B is reused for the family in the statement. Clarify the distinction between the subvariety of G and its intersection with P_U(E).","section":"Theorem 7.5 proof"}],"recommendation":"reject","confidential_remarks":"The paper contains a substantial and interesting framework in Sections 3–6, and several of the local results appear sound. However, the central existence theorem for the pencil of plane quartics depends on Lemma 7.3, which is false. The proof of Theorem 7.5, and hence the application in Theorem 7.8, is therefore invalid. This is not a local fixable detail: the mechanism forcing the required cone to have vertex in the open set U is missing. Unless the authors can supply a new argument that avoids the false lemma, the main claim is unproven. I recommend rejection, though I would encourage the authors to resubmit a version with the Section 3–6 material separated from the unsupported application."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing you need to know: this paper has a genuinely new framework, but the main application is not proved. Theorem 7.8 goes through Theorem 7.5, and the only step forcing a cone with vertex in the open set U into the inverse image of 16q is Lemma 7.3. That lemma's proof is wrong: it says a proper subvariety of an affine open must be zero-dimensional, which is simply false. The lemma might be true by a semicontinuity argument, but the text gives no such argument. The stress-test counterexample that was floated does not convince me — the blow-up example still has the strict transform mapping onto P^1, and the multiplication map on P^1×P^1 is not regular — but that is beside the point: the proof in the paper is invalid, and the theorem is unsupported.\n\nWhat is actually new and good: the systematic treatment of linear limits of cones, the cone map to the Grassmannian and its partial extension over the blow-up, and the differential analysis of Theorem 6.8. The degree-4 case (elliptic normal curve) is worked out with real geometric content, and the connection to the earlier cubic paper is motivational, not load-bearing. Sections 3-6 look like solid, careful algebraic geometry. The authors are not fitting parameters or hiding circular reasoning; the flaw is a missing proof step in one lemma, not a fake result.\n\nThe soft spot is proportional: it is concentrated in Section 7. Lemma 7.3 is used exactly once, to pass from a nonempty preimage of D to a preimage that meets the open cone locus. Without that, Theorem 7.5 cannot be established, and Theorem 7.8 has no certified cone to project from. The rest of the pencil argument — irreducibility, smoothness, non-isotriviality — is conditional on having that one cone. So the advertised result is not currently proven.\n\nWho this is for: specialists in linear series on space curves and classical projective geometry. The framework deserves serious referee time; the existence theorem should not be accepted in current form. I would send this to a competent referee and expect major revision: fix or replace Lemma 7.3, then confirm Theorem 7.5 survives. It is a desk-rejectable paper only if you think the whole framework is worthless, which it is not.","headline":"The conic-linear-series framework is substantial, but the advertised pencil of quartics rests on a lemma whose proof is invalid, so the main theorem is unsupported as written.","tokens_in":26151,"tokens_out":18849,"would_cite":false,"duration_ms":203433,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H50","14N05","14C20","14H52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs a non-isotrivial pencil of plane quartic curves with exactly one base point, all members irreducible and the general member smooth.","keywords":["conic linear systems","plane quartics","pencils with one base point","non-isotrivial pencils","elliptic normal curves","linear limits of cones","cone map"],"falsifier":"A direct test of the load-bearing step: check whether for a specific elliptic quartic and a 16-torsion point q the inverse image (Φ'')^{-1}(16q) actually meets the open cone locus P_U(E). The proof forces this by Lemma 7.3, but that lemma has a concrete counterexample: X=P^1×P^1 with φ the first projection and μ([s:t],[u:v])=[su:tv] maps every general φ-fiber onto P^1 while the fiber over [1:0] maps to a single point.","tokens_in":25095,"feed_emoji":"📐","tokens_out":8633,"duration_ms":82053,"temperature":0.7,"pith_summary":"The paper develops a theory of conic linear series: linear systems on a smooth complex curve C ⊂ P^3 cut out by cones of fixed degree with vertex outside C, together with their limits as the vertex moves onto the curve. Using this theory, it proves that a non-isotrivial pencil of plane quartics can have all its members irreducible, a smooth general member, and a base locus reduced to a single point. The road to this result is the claim that for an elliptic normal quartic, every effective divisor of degree 16 is cut by a one-dimensional family of quartic cones; projecting one such cone together with the curve from the cone's vertex yields the desired pencil. This extends a known construction for plane cubics to degree four.","feed_headline":"A quartic pencil with exactly one base point is shown to exist","feed_subtitle":"All members irreducible, general member smooth, and the pencil is non-isotrivial — the quartic analog of the cubic case.","key_machinery":"The central mechanism is the cone map ρ_d : U → G(n_d, S_d), which sends a point p to the linear system cut out on C by cones of degree d with vertex p. The paper proves that this map extends to the blow-up of P^3 along C via 'linear limits of cones': when the vertex specializes to p ∈ C along a line ℓ not tangent to C, the limiting degree-d cone is the cone from p over C together with the plane spanned by ℓ and the tangent line at p. This extension yields a morphism Φ : P_U(E) → P(S_d) whose differential has generically maximal rank, with kernel dimension 1 exactly for (d,g)=(4,1); the resulting dominance of Φ for d ≤ 4 is what forces the existence of the one-parameter families of cones cut","core_discovery":"On a smooth degree-4 genus-1 curve C ⊂ P^3, the conic map Φ from the projectivized cone bundle P_U(E) to the complete linear system P(S_4) of degree-16 divisors is dominant with one-dimensional kernel, and each effective degree-16 divisor D is claimed to be cut out by a one-dimensional family of cones whose vertices trace a curve in the open set U. Applying this to a point q with 16q ∼ 16O but 8q ≁ 8O, the paper takes one such cone, projects from its vertex along with C, and obtains a pencil of plane quartics whose base locus is exactly the projection of q; irreducibility follows from the 8q condition, smoothness of the general member from an analysis of the inverse image of 16q, and non-iso","pith_inferences":["The same linear-limit machinery could be carried to curves in P^n for n ≥ 4, replacing cones by hypersurfaces with prescribed vertex; if the dominant-range result generalizes, one could produce analogous pencils of hypersurfaces with one base point.","The one-dimensional kernel in the elliptic quartic case suggests the inverse images of divisors under Φ carry a natural foliation structure on P^2 minus a point, a direction the paper hints at in connection with degree-4 foliations.","The evaluation-at-least-(d-2) property of limit conic linear series gives a new way to spot Weierstrass-type filtrations on a curve, potentially linking conic limits to higher-order Weierstrass points."],"forward_implications":["There exists a non-isotrivial pencil of plane quartics with exactly one base point, all members irreducible, and general member smooth.","Every effective degree-16 divisor on an elliptic normal quartic is cut out by a one-dimensional family of quartic cones with vertices in the open set U.","The dominance of the cone map holds precisely for degree d ≤ 4; for d ≥ 5 conic divisors form a proper subvariety of the complete linear system.","The differential of the cone map has kernel dimension 0, 1, or 2 according to (d,g): injective for d ≥ 5 or (d,g)=(4,0); one-dimensional for (4,1); two-dimensional for (3,0)."],"fun_headline_variants":["One base point, irreducible fibers: a rigid quartic pencil","Quartic pencil with a unique base point and smooth general member","Non-isotrivial pencil of quartics: one base point, all irreducible","Solitary base point anchors a non-splitting quartic pencil"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction of the pencil rests on the claim that if the inverse image of a fixed divisor under the extended cone map projects to a positive-dimensional set for the general divisor, it must do so for every divisor — a semicontinuity assertion (Lemma 7.3) that fails for simple maps between surfaces.","fun_headline_variants_meta":{"raw":{"variants":["One base point, irreducible fibers: a rigid quartic pencil","Quartic pencil with a unique base point and smooth general member","Non-isotrivial pencil of quartics: one base point, all irreducible","Solitary base point anchors a non-splitting quartic pencil"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000209,"raw_usage":{"total_tokens":1183,"prompt_tokens":625,"completion_tokens":558,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":369,"completion_tokens_details":{"reasoning_tokens":482}},"tokens_in":369,"tokens_out":558,"duration_ms":5898,"temperature":1.0,"reasoning_tokens":482,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T22:27:38.045987+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test of the load-bearing step: check whether for a specific elliptic quartic and a 16-torsion point q the inverse image (Φ'')^{-1}(16q) actually meets the open cone locus P_U(E). The proof forces this by Lemma 7.3, but that lemma has a concrete counterexample: X=P^1×P^1 with φ the first projection and μ([s:t],[u:v])=[su:tv] maps every general φ-fiber onto P^1 while the fiber over [1:0] maps to a single point.","supporting_citations":[],"review_version":1}