{"id":"3b291ea5-a04a-420b-90a7-b8c1d54d70ef","arxiv_id":"2507.13280","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Hirzebruch surfaces, the algebraic exceptional set of a three-component normal-crossing curve with big K+B and no fiber or negative self-intersection component is finite and equals the hyper-bitangent set.","lead":"This paper proves that on a Hirzebruch surface, a three-component normal-crossing curve B with big K+B and no fiber or negative self-intersection component has a finite algebraic exceptional set, and that this set equals the set of hyper-bitangent curves. It is a new case of a conjecture linking algebraic hyperbolicity with Diophantine approximation, with effective bounds in most cases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The finiteness proof for the F1 subcase {B1,B2}⊂|C1| in Theorem 4.6(b) rests on an unverified application of [9, Thm.1]: the paper does not show that P^2\\π(B) is a finite affine smooth cover of G_m^2 satisfying the hypotheses of [9], and the image divisor may fail to be SNC at q=π(C0).","rationale":"The paper's central theorem extends Caporaso–Turchet to Hirzebruch surfaces. Most of the proof is elementary and internally consistent: the effective bounds for F0 and Fe with e≥2, the equality E(B)=Hyp(B,2) on F1, and the finiteness in Theorem 4.6(a) do not depend on the contested step. The single load-bearing external input is the invocation of [9, Thm.1] in Theorem 4.6(b) to bound d1+1 by γ(B). The reader's weakest-assumption analysis identifies exactly this point, and my reading confirms that the paper supplies no verification that P^2\\π(B) satisfies the hypotheses of [9]. There is a plausible way to fill the gap using the map [l1^b:l2^b:F3], but the possible non-SNC behaviour of π(B) at q=π(C0) makes the transfer non-obvious and potentially delicate, especially since π(D) itself passes through q. Because the gap affects only the finiteness assertion in one F1 subcase and not the equality statement or the other cases, conditional acceptance is the appropriate verdict: the concern is real and must be settled, but it does not warrant rejection unless the required hypothesis check is shown to be impossible.","tokens_in":30829,"tokens_out":21147,"duration_ms":246628,"concrete_test":"Verify the applicability of [9, Thm.1] to (P2,π(B)) in the {B1,B2}⊂|C1| case: write l1,l2,F3 for the equations of π(B1),π(B2),π(B3), construct the finite morphism Φ=[l1^{deg F3}:l2^{deg F3}:F3] from P^2\\π(B) to G_m^2, and check the exact hypotheses of [9, Thm.1], especially log-smoothness at q=π(C0) when β3≥2 or B3 is tangent to C0. If [9] requires an SNC compactification, repeat the argument on a log resolution and prove that the degree bound transfers to deg π(D); if it does not transfer, Theorem 4.6(b) must be revised and the finiteness claim in that subcase is not proved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Theorem 4.6(b), after contracting C0, the paper asserts that “π(B) is still a 3C-curve” and immediately applies the Corvaja–Zannier theorem [9, Thm.1] to conclude deg π(D)=d1+1≤γ(B). No hypothesis of [9, Thm.1] is verified in the text. A finite morphism from P^2\\π(B) to G_m^2 can probably be written as [l1^b:l2^b:F3] when B3 avoids p12=B1∩B2, but this construction is not given. More seriously, when β3≥2 or B3 is tangent to C0, the plane curve F3=π(B3) has a singular point at q=π(C0), so the divisor π(B) is not simple normal crossing at q. Whether [9, Thm.1] permits such boundary singularities is exactly the missing check. If it requires an SNC compactification, the bound d1+1≤γ(B) is unsupported and the finiteness of E(B) in this F1 subcase is not established, although the equality E(B)=Hyp(B,2) and the effective bounds in all other cases are independent of this step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the algebraic exceptional set E(B) of a reduced curve B with three irreducible components and normal crossings at component intersections on a Hirzebruch surface F_e. Under the assumptions that K_{F_e}+B is big and that no component of B is a fiber or the negative section C0, the paper claims that E(B) equals the set Hyp(B,2) of curves meeting B in at most two points after normalization, and that E(B) is finite. The proof is case-by-case: F0 and F_e (e≥2) are handled in Section 3 with explicit enumerative bounds; the remaining surface F1 is treated in Section 4, where the equality is proved and finiteness is established by reduction to P2 results and, in one subcase, by an appeal to a theorem of Corvaja–Zannier. The paper also gives effective bounds for |E(B)| in most cases and shows that E(B)=∅ for general B under additional numerical assumptions.","tokens_in":31000,"tokens_out":20652,"duration_ms":207107,"significance":"If the proof is correct, the paper gives the first extension beyond P2 of the finiteness and equality results for the algebraic exceptional set of three-component curves, confirming Lang's conjecture in this class of log-general-type surfaces. The paper is carefully written and contains genuinely useful local tools: Theorem 2.11 (a δ-invariant lower bound at unibranch tangencies) and the systematic use of the strong triangle inequality (Theorem 2.13) are stated and proved in detail. The effective finiteness bounds in Sections 3 and most of Section 4 are obtained by explicit counting, which is a clear strength. The main caveat is the application of the Corvaja–Zannier theorem in Theorem 4.6(b), where the hypotheses on P^2\\π(B) are not verified; if that gap can be closed, the result is a substantial confirmation of the expected picture.","major_comments":[{"comment":"The application of the Corvaja–Zannier theorem [9, Thm. 1] is not justified. The proof asserts that after contracting C0 the divisor π(B) is still a 3C-curve and that the Corvaja–Zannier theorem gives d1+1 = deg π(D) ≤ γ(B), but none of the hypotheses of [9, Thm. 1] is verified. In particular, the paper does not construct a finite morphism P2\\π(B) → G_m^2, and it does not check that the boundary π(B) satisfies the regularity conditions required by [9] (for instance, simple normal crossing support). This is not a technicality: under the standing assumptions of Prop. 4.1 we have β3 ≥ 1, so q = π(C0) lies on π(B3); unless β3 = 1 and B3 meets C0 transversally, the curve π(B3) is singular at q, and even in the smooth case the paper gives no argument that [9] permits this configuration. Since the bound deg π(D) ≤ γ(B) is the only input that prevents the set of degrees {d1} from being infinite, the finiteness of E(B) in this subcase—and hence part (2) of Theorem 1 for F1—is not established by the present argument. The equality E(B) = Hyp(B,2) and the effective bounds in all other cases are independent of this step.","section":"§4, Theorem 4.6(b)"}],"minor_comments":[{"comment":"Remark 3.9 states that 'the cases we study are not covered by [4],[8], and [9]', which appears to be in tension with the central use of [9, Thm. 1] in Theorem 4.6(b). Please clarify the intended scope, for example by distinguishing F_e with e≥2 from the F1 subcase, and by indicating that the application of [9] is to the contracted plane complement P2\\π(B) rather than to F_e\\B.","section":"Remark 3.9"},{"comment":"The reduction to [4] would benefit from a short verification that π(B) is a 3C-curve on P2 and that the hypotheses of the cited results from [4] are satisfied; the current text says only that 'π(B) is still a 3C-curve'.","section":"Prop. 4.3(2)"},{"comment":"Lemma 2.19 is cited to a MathOverflow answer; if a standard textbook statement is available (for example, [14, Ch. V]), it would be more suitable for a journal article.","section":"Lemma 2.19"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is largely sound and well structured, and the local results in Section 2 are valuable in their own right. The one point I would ask the editor to weigh is the unverified use of the Corvaja–Zannier theorem in Theorem 4.6(b); I would not accept the paper until that application is either justified or replaced by an alternative argument. The authors should also clarify the apparent tension between Remark 3.9 and the use of [9] in Section 4."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper settles Lang's algebraic exceptional set conjecture for three-component normal-crossing boundaries on all Hirzebruch surfaces, provided no component is a fiber or the negative section. That is a real extension of Caporaso–Turchet's P^2 theorem and the first such result on surfaces other than P^2. The proof is mostly careful and case-by-case, with effective bounds for every case except one subcase.\n\nWhat is good: the generalized local inequality Thm 2.11 is a useful addition, the counting arguments are precise, and the paper is honest about where its method stops (Rem 3.9, Example 4.10). The reduction to [4] in Section 4 for β=0 is sound. The equality E(B)=Hyp(B,2) is shown for F0, Fe (e≥2), and F1, and does not depend on the questionable step.\n\nThe soft spot is exactly the one you'd spot: Theorem 4.6(b). When B1,B2∈|C1|, the proof contracts C0 and applies Corvaja–Zannier [9, Thm.1] to get deg π(D)≤γ(B). But the hypotheses of that theorem are not verified. The divisor π(B) generally has a singular point at q=π(C0) when β3≥2 or when B3 is tangent to C0, so it is not SNC there; the finite morphism to G_m^2 is not exhibited. As written, the bound is unsupported. This only affects finiteness in that subcase; the equality and the other effective bounds stand on their own.\n\nI think the gap is fixable, not fatal. The complement of two lines and a curve on P^2 most likely does admit a finite map to G_m^2, but the paper needs to show it and check the hypotheses. A referee should request this.\n\nRecommendation: send to peer review. It deserves referee time. The author will have to fill the C-Z gap before the full Theorem 1 is airtight.","headline":"A genuine extension of Caporaso–Turchet to all Hirzebruch surfaces, with one F1 subcase where finiteness rests on an unverified Corvaja–Zannier application.","tokens_in":31651,"tokens_out":5362,"would_cite":true,"duration_ms":63740,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H20","14H45","14J26"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a three-component normal-crossing curve B on any Hirzebruch surface with K+B big, the paper proves the algebraic exceptional set equals the hyper-bitangent curves and is finite, with effective bounds in all but one F1 case.","keywords":["algebraic exceptional set","Hirzebruch surfaces","hyper-bitangent curves","hypertangency","three-component curves","rational curves","log-general type","unibranch points"],"falsifier":"Take B1,B2∈|C1| on F1 and B3∈|C1+f| in general position, and look for integral curves D∈|dC1+f| with #$ν_D^{{-1}}$(D∩B)≤2 for arbitrarily large d. The paper's proof bounds such degrees only through the external theorem [9]; finding infinitely many such curves would refute Theorem 1, while checking whether $P^{2}$\\π(B) satisfies the hypotheses of [9] would directly test the proof's missing step.","tokens_in":30496,"feed_emoji":"📐","tokens_out":10953,"duration_ms":112468,"temperature":0.7,"pith_summary":"The paper studies the algebraic exceptional set of a curve B with three irreducible components meeting normally on a Hirzebruch surface F_e: the set of rational curves that meet B in at most two points after normalization. It proves that, when K_{F_e}+B is big and no component of B is a ruling fiber or the negative self-intersection section, this exceptional set coincides with the set of hyper-bitangent curves and is finite. This confirms the conjectural finiteness statement for every such three-component log-smooth pair on a Hirzebruch surface, extending a previously known plane-curve result to the full family. The proof yields explicit bounds on the number of exceptional curves in all but one subcase on F_1, where it invokes an external degree bound.","feed_headline":"Exceptional set is finite for 3-component curves on Hirzebruch surfaces","feed_subtitle":"Extends the plane-curve theorem: exceptional set equals hyper-bitangent curves and is finite, with effective bounds in most cases.","key_machinery":"The central object is Hyp(B,2), the set of hyper-bitangent curves: integral curves D with #$ν_D^{{-1}}$(D∩B)≤2, meaning that after normalization the preimage of B has at most two points. Two local statements carry the argument. Theorem 2.11 lower-bounds the δ-invariant of a singular unibranch point of D (a local contribution to arithmetic genus) in terms of its multiplicity and its intersection multiplicity with a smooth branch of B. Theorem 2.13, the strong triangle inequality, says that when three curves share a point that is unibranch on all of them, the smallest two normalized intersection multiplicities are equal; the paper uses this to rule out two distinct curves in the same linear system through the same two points. Together these tools reduce the classification of hyper-bitangent curves to a short list of divisor classes and turn finiteness into an explicit count, except in one F_1 subcase where a degree bound from [9] is imported.","core_discovery":"On its own terms, the paper's central claim is Theorem 1: if S is a Hirzebruch surface and B is a 3C-curve (three irreducible components, normal crossings at every intersection point) such that K_S+B is big and no component lies in |f| or equals C_0, then E(B)=Hyp(B,2) and E(B) is finite. The equality is established by classifying possible divisor classes of a curve D counted by Hyp(B,2): on F_0 the only classes are |(1,0)|, |(0,1)|, and |(1,1)|; on F_e with e≥2 they are |C_0|, |f|, and, only when e=2, |C_1|; on F_1 they are |C_0|, |f|, |C_1|, and |dC_1+f| with d≥1. Once the class is fixed, arithmetic genus computations force D to be rational, and finiteness follows by counting curves through the finitely many intersection points of B's components, with the strong triangle inequality used to show at most one curve passes through a given pair of points.","pith_inferences":["Inference: the two local estimates (Theorem 2.11 and the strong triangle inequality) are not special to Hirzebruch surfaces, so the same classification strategy may carry over to other toric or rational surfaces; this is not claimed in the paper.","Inference: Example 4.10 suggests the hypotheses forbidding fibers and the negative section are close to sharp, since dropping them allows hyper-bitangent curves of positive genus; one testable consequence is that the equality E(B)=Hyp(B,2) should fail exactly when a component is a fiber or C_0.","Inference: the unverified applicability of [9] in the F_1 subcase could be checked by direct computation for small examples, and a positive check would turn the only non-effective part of the theorem into an effective one.","Inference: the effective bounds, being polynomial in the intersection numbers of the components, suggest a natural quantitative strengthening in which the number of exceptional curves is bounded by a universal function of the self-intersection data, independent of the position of B."],"forward_implications":["Lang's finiteness conjecture for algebraic exceptional sets holds for every three-component normal-crossing boundary on every Hirzebruch surface under the stated hypotheses.","The equality E(B)=Hyp(B,2) means that any integral curve meeting B in at most two normalization-preimage points must be rational, so the exceptional set can be tested by looking only at hyper-bitangent curves.","The bounds are effective in all but one case: for example, on F_0 the exceptional set has at most 24 elements when all three components are of type (1,1), and on F_2 it has at most 19 when all three are of type |C_1|.","For general B of sufficiently large degree, the exceptional set is empty: no curve is hyper-bitangent (Propositions 3.4, 3.8, and 4.9).","On F_1, hyper-bitangent curves occur only in the divisor classes |C_0|, |f|, |C_1|, and |dC_1+f|, and for dC_1 with d≥3 only when two components lie in |C_1|."],"supporting_citations":[{"why":"Supplies the definition of hyper-bitangent curves and the plane-case equality E(P^2,B)=Hyp(P^2,B,2) that this paper extends to Hirzebruch surfaces.","marker":"[4]"},{"why":"Provides the degree bound used in Theorem 4.6(b) to prove finiteness in the F1 subcase where two components lie in |C1|.","marker":"[9]"},{"why":"Gives the finiteness theorem for three-component curves in P^2 that this paper generalizes, and the explicit bound used in Example 4.8.","marker":"[8]"},{"why":"Provides the multiplicity-sequence lemma (Lemma 2.6) controlling intersection multiplicities at unibranch points.","marker":"[10]"},{"why":"Provides the strong triangle inequality used to force uniqueness of curves through two fixed intersection points.","marker":"[12]"},{"why":"Sets out the Picard group, divisor classes, intersection numbers, and bigness criterion for Hirzebruch surfaces used throughout.","marker":"[14]"}],"fun_headline_variants":["Finite exceptional set: 3-component curves on Hirzebruch surfaces","Exceptional set = hyper-bitangents, finite for 3-part curves","Effective bounds for algebraic exceptional sets on Hirzebruch surfaces","3-component curves: exceptional set finite, hyper-bitangent","Exceptional set equals hyper-bitangents: finite for Hirzebruch 3-curves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"In the special F1 case where two components lie in |C1|, the proof of finiteness relies on applying an external degree bound to the plane curve obtained by contracting the negative section, and the paper does not check that this plane curve satisfies the conditions the external bound requires.","fun_headline_variants_meta":{"raw":{"variants":["Finite exceptional set: 3-component curves on Hirzebruch surfaces","Exceptional set = hyper-bitangents, finite for 3-part curves","Effective bounds for algebraic exceptional sets on Hirzebruch surfaces","3-component curves: exceptional set finite, hyper-bitangent","Exceptional set equals hyper-bitangents: finite for Hirzebruch 3-curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000581,"raw_usage":{"total_tokens":2707,"prompt_tokens":887,"completion_tokens":1820,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":1721}},"tokens_in":503,"tokens_out":1820,"duration_ms":12949,"temperature":1.0,"reasoning_tokens":1721,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:29:11.299442+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take B1,B2∈|C1| on F1 and B3∈|C1+f| in general position, and look for integral curves D∈|dC1+f| with #$ν_D^{{-1}}$(D∩B)≤2 for arbitrarily large d. The paper's proof bounds such degrees only through the external theorem [9]; finding infinitely many such curves would refute Theorem 1, while checking whether $P^{2}$\\π(B) satisfies the hypotheses of [9] would directly test the proof's missing step.","supporting_citations":[{"cited_title":"Hypertangency of plane curves and the algebraic ex- ceptional set.Proceedings of the London Mathematical Society, 130(6):e70063, 2025","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of hyper-bitangent curves and the plane-case equality E(P^2,B)=Hyp(P^2,B,2) that this paper extends to Hirzebruch surfaces."},{"cited_title":"Algebraic hyperbolicity of ramified covers ofG2 m (and integral points on affine subsets ofP2).J","cited_arxiv_id":null,"evidence_quote":"Provides the degree bound used in Theorem 4.6(b) to prove finiteness in the F1 subcase where two components lie in |C1|."},{"cited_title":"Some cases of Vojta’s conjecture on integral points over function fields.J","cited_arxiv_id":null,"evidence_quote":"Gives the finiteness theorem for three-component curves in P^2 that this paper generalizes, and the explicit bound used in Example 4.8."},{"cited_title":"On a class of rational cuspidal plane curves","cited_arxiv_id":null,"evidence_quote":"Provides the multiplicity-sequence lemma (Lemma 2.6) controlling intersection multiplicities at unibranch points."},{"cited_title":"An approach to plane algebroid branches","cited_arxiv_id":null,"evidence_quote":"Provides the strong triangle inequality used to force uniqueness of curves through two fixed intersection points."}],"review_version":1}