{"id":"d7f24a45-16dd-41bc-a46b-75cb97676f25","arxiv_id":"1908.04706","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For simple normal crossings divisors, logarithmic and local/naive Gromov-Witten invariants with maximal contacts differ, and this paper gives the first counterexamples plus a blowup formula measuring the difference.","lead":"This paper disproves two conjectures saying that two ways of counting rational curves with tangency conditions always agree, then quantifies exactly where and how they differ. The difference is controlled by an explicit blowup construction governed by tropical geometry, and a positive case is proven for products of projective spaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1.3's four-line component is also certified by the same numerical criterion for the two-divisor moduli; Prop 1.2's inequality is therefore not established.","rationale":"The paper's headline result (Theorem X) is a negative answer to the van Garrel–Graber–Ruddat and Tseng–You conjectures, so its proof must genuinely separate the naive intersection from the two-pointed logarithmic class. The reader's weakest assumption concerns unproved dimension bounds in Proposition 1.2, but the concern identified here is more direct: even if every asserted dimension is correct, the component analysis does not distinguish the two sides. The numerical criterion used in Lemma 1.3 is the paper's own tool, and applying it to the two-divisor moduli space certifies the same four-line type, or at least the paper gives no reason it should fail there. This makes the claimed inequality unsupported: the two sides could share both the main component and the four-line component, with the inequality depending on uncomputed multiplicities. The degree-2 case in Section 1.2 has the same structure. Because the central disproof is not established as written, the verdict should move from CONDITIONAL to UNVERDICTED rather than REJECT: the counterexample might still be true with additional work, but the proof given is incomplete. The correction machinery of Theorems Y, Z, and Theorem W are independent contributions and are not affected by this specific gap, which is why a full rejection is not warranted.","tokens_in":23062,"tokens_out":37727,"duration_ms":408644,"concrete_test":"Apply Gathmann's numerical balancing criterion [12, Remark 1.7(ii)] to the four-line combinatorial type of Lemma 1.3 inside the two-pointed space Kmax_0,2(P2|H1+H2,4). For each line component, the sum of contact orders at its unique node is (1,1), which equals its intersection with H1+H2, and the contracted component C0 maps into D, so the criterion certifies the type. If this certification holds, determine whether the four-line locus is contained in π(Kmax_0,2(P2|H1+H2,4)); if yes, compute its multiplicity on both sides of equation (1) via the degeneration/tropical formula, and re-evaluate Proposition 1.2. The same test should be repeated for the two-line locus in Section 1.2.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The counterexample to the original form of the correspondence rests entirely on Proposition 1.2, whose proof relies on Lemma 1.3 exhibiting a 'second irreducible component' of the intersection (2): four lines through p=H1∩H2 together with a contracted component C0 carrying the marking. Lemma 1.3 places this locus in each image πi(Kmax_0,1(P2|Hi,4)) by Gathmann's numerical balancing criterion [12, Remark 1.7(ii)]: each non-contracted line has degree 1 along Hi and one node of contact order 1, while C0 maps into Hi and is therefore unconstrained. But the identical computation applies verbatim to the two-pointed space Kmax_0,2(P2|H1+H2,4): each line meets H1+H2 with degree 2 and carries one node with contact orders (1,1), and C0 maps into D, so the criterion again certifies the type. Thus the same argument that puts the four-line locus on the left-hand side of (1) also puts it on the right-hand side, in π(Kmax_0,2(P2|H1+H2,4)). If so, this locus is not an excess component of the left-hand side relative to the right-hand side, and the final sentence of the proof of Proposition 1.2 — 'the component [π(Kmax...)] appears on both sides, so the two sides cannot be equal' — is invalid: both sides may contain both components, and no multiplicities are computed. The same gap affects the degree-2 pointed counterexample in Section 1.2, where the two-line-plus-contracted-component locus is equally certified by the criterion for the two-divisor space. The paper never proves that the second component is absent from the two-divisor image, nor that it lies outside the closure of the main component; both are essential for the claimed inequality.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies genus-zero logarithmic Gromov-Witten invariants with maximal contact orders along simple normal crossings divisors. It introduces a \"naive\" theory obtained by intersecting the relative maximal-contact loci for the individual divisor components and observes that this naive theory agrees with the local theory up to signs and pushforwards. The central claim is that the logarithmic and naive/local theories do not coincide in general: Theorem X asserts that for P^2 with the divisor H1+H2 of two lines, the strong form of the local/logarithmic correspondence fails in degree 2 and the original form fails in degree 4. The paper then proposes a correction mechanism via an explicit sequence of weighted blowups of the Kontsevich space (Theorems Y, Z and 3.6), expressing the difference between the logarithmic and naive classes as a sum of tautological correction terms, and proves a positive result (Theorem W) for product geometries with factorwise primary insertions.","tokens_in":23181,"tokens_out":16997,"duration_ms":188501,"significance":"If the counterexamples in Theorem X are correct, they would settle negatively the van Garrel-Graber-Ruddat and Tseng-You local/logarithmic conjectures, which had substantial numerical support. The paper also contains a novel and potentially very useful algorithmic description of the relevant birational modifications in terms of tropical image-orderings and floral strata, and the corrected product formula is a concrete quantitative replacement for the failed correspondence. These positive contributions are substantial regardless of the fate of the counterexamples. However, the counterexamples are the paper's headline result, and their proof as written has load-bearing gaps.","major_comments":[{"comment":"The proof of Proposition 1.2 depends on the assertion that the four-line locus described in Lemma 1.3 is an irreducible component of the intersection (2) that is not present on the right-hand side. The paper never proves this. The numerical balancing criterion [12, Remark 1.7(ii)] is used to show that the four-line locus lies in each of π1(Kmax_0,1(P2|H1,4)) and π2(Kmax_0,1(P2|H2,4)); the same computation appears to certify the same locus for π(Kmax_0,2(P2|H1+H2,4)), since each non-contracted line meets H1+H2 with degree 2 and carries one node with contact orders (1,1), while C0 maps into D. If this is so, the four-line locus is contained in the main component on the right-hand side rather than being an excess component, and the final sentence of the proof of Proposition 1.2 — that the component [π(Kmax...)] appears on both sides, so the two sides cannot be equal — is invalid. The paper must either prove, using the contact-order conditions at the marked points or another mechanism, that the four-line locus is not in the image of Kmax_0,2(P2|H1+H2,4), or compute the multiplicities with which it appears on each side.","section":"§1.1, proof of Proposition 1.2"},{"comment":"The claim that every irreducible component of the intersection (2) has dimension exactly 5 is not established. The boundary-stratum analysis is asserted: a boundary stratum of Kmax_0,1(P2|Hi,4) has dimension at most 7, forgetting the marking reduces the dimension to 6 unless the marking lies on a contracted tail, and the remaining conic-plus-two-lines case is bounded by \"elementary geometry\". No proof or reference for these bounds is supplied, and the phrase \"cuts the dimension to at least 5\" is at best ambiguous and at worst contradicts the intended \"at most 5\". This is load-bearing because [11, Proposition 7.1] is invoked to decompose the left-hand side as a positive sum of component classes, which requires a proper intersection; a boundary contribution of dimension 6 would invalidate the inequality in Proposition 1.2.","section":"§1.2, pointed counterexample"},{"comment":"The proof of the inequality (3) for the degree-2 pointed counterexample is incomplete. The text says \"We leave some verifications to the reader\" and \"Direct analysis shows that there are no further irreducible components\", but this analysis is the core of a central counterexample. In particular, the argument does not show that the second 3-dimensional component is not contained in Kmax_0,2(P2|H1+H2,2), and no multiplicities are computed for either component. The same concern as in the degree-4 case applies: if the two-line-plus-contracted-component locus is also certified by the numerical criterion for the two-divisor moduli space, then it is not an excess component and the inequality does not follow.","section":"§1.2, proof of (3)"}],"minor_comments":[{"comment":"The sentence \"which cuts the dimension to at least 5\" should presumably read \"at most 5\"; as written it undermines the dimension bound it is meant to establish.","section":"§0.3"},{"comment":"The sentence \"We do not not extract a closed form solution\" contains a typo: \"do not not\" should be \"do not\".","section":"§2.6"},{"comment":"In the proof of Theorem 2.13, the transition from the target edge lengths to the flower subcone (9) is terse; a short explanation of why the degree of the root changes from β0 to β0+β1 would improve readability.","section":"§1.1, Lemma 1.1"},{"comment":"The factor 42 in Lemma 1.1 is introduced without comment; a one-sentence derivation (for instance, from the product of the two evaluation degrees) would help the reader check the normalization.","section":"§5.2, Theorem 5.2"},{"comment":"The proof of Theorem 5.2 says the result follows by a diagram chase; the compatibility of the local classes with the birational morphism θ is plausible but should be stated more explicitly, since it is the key input.","section":"References"},{"comment":"Reference [18] contains a typographical artifact \"Pr\"; it should be \"P^r\".","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's main positive framework — the explicit blowup sequence and the corrected product formula — is interesting and likely important, but the headline counterexamples are not proven as written. The key risk is in Section 1: the proof of Proposition 1.2 does not rule out the possibility that the alleged excess component also lies on the right-hand side, and the dimension bound for boundary contributions is asserted rather than proved. If the authors can supply a rigorous proof of the counterexamples, or alternatively remove the overclaim and reframe the paper around the correction machinery and the product case, the contribution would be solid. I would not accept the paper in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know first: this is a serious attack on the local/logarithmic correspondence, with real new content. The claimed counterexamples to the van Garrel–Graber–Ruddat and Tseng–You conjectures, if correct, are the first that reverse conjectures with numerical support. The blowup construction (Theorem Z) is a genuine advance: explicit, algorithmic, and canonical, unlike earlier non-explicit birational modifications. The product-geometry theorem (Theorem W) is elementary but useful, giving the first non-toric examples in dimension >2. The authors also honestly disclose that an earlier version had a load-bearing error and describe the Fulton blowup-formula fix; that is the right structural move.\n\nThe main soft spot is the proof of Proposition 1.2, the unpointed counterexample. The dimension bound for boundary contributions is asserted (\"elementary geometry\"), and the pointed degree-2 case explicitly leaves verifications to the reader. A stress-test worry that the four-line component is also certified for the two-divisor moduli seems unlikely on dimension grounds: the two-divisor moduli is 5-dimensional, and the four-line locus would give a 7-dimensional stratum there, so it cannot be an irreducible component of its image. But the paper never explicitly rules out that the component appears on both sides, and the final sentence of the proof is too fast. That needs a real argument or a reference, not an assertion.\n\nThe abstract claims the difference is \"explicitly determined,\" but the concrete correction terms are deferred to ongoing work (Remark 3.7); what is given is a structural formula with tautological excess classes. The constant 42 in Lemma 1.1 is stated without derivation and sits oddly next to the factor 16 in Lemma 3.1; a short explanation would remove the worry. These are reparable issues, not obviously fatal.\n\nThis paper deserves a serious referee. The central claims are important and the machinery is likely to be influential even if the counterexample proof needs tightening. I would send it to review, with a request that the authors fill the dimension arguments and soften the abstract's explicitness claims.","headline":"A serious paper with two genuine new ideas — counterexamples to the local/logarithmic conjectures and an explicit blowup correction — but the counterexample proof has real gaps and the abstract oversells the explicitness.","tokens_in":23979,"tokens_out":24501,"would_cite":true,"duration_ms":263811,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14N10","14T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Maximal-contact Gromov–Witten invariants do not agree with the naive or local theories in the plane with two lines, and the paper computes the difference via weighted blowups.","keywords":["logarithmic Gromov-Witten theory","maximal contact","local Gromov-Witten invariants","relative Gromov-Witten theory","tropical moduli of maps","weighted blowups","simple normal crossings divisor","enumerative geometry"],"falsifier":"Check the dimension of the boundary locus in $K_{0,0}(\\mathbb{P}^2,4)$ consisting of maps whose domain is a contracted component at $H_1\\cap H_2$ glued to four rational tails mapping isomorphically onto lines. Proposition 1.2 requires this locus to have dimension 5; an explicit parametrization (for instance by gluing parameters or equivariant localization) that produced dimension 6 would make the intersection improper and break the claimed inequality, while a clean dimension-5 computation would confirm the counterexample.","tokens_in":22638,"feed_emoji":"📐","tokens_out":21288,"duration_ms":193770,"temperature":0.7,"pith_summary":"The paper compares three ways of counting genus-zero rational curves that meet a simple normal crossings divisor with maximal order at the marked points: the logarithmic theory, built from logarithmic structures; the naive theory, whose classes are the products of the relative Gromov–Witten classes of the individual divisor components; and the local theory, which counts curves in the total space of the sum of the normal line bundles. Its central claim is that the logarithmic and naive/local theories need not coincide, and it demonstrates this with explicit counterexamples in the projective plane with two lines as divisor: the strong form of the correspondence fails in degree 2 and the original form, obtained by forgetting the marked points, fails in degree 4. The failure comes from extra components in the intersection of the two relative loci, namely curves whose domain has a contracted component at the intersection point of the two lines. The paper replaces the conjectured identity with a correction formula in which the difference is captured by an explicit sequence of weighted blowups along combinatorial 'floral' strata, and it proves that for product geometries the correspondence holds with primary factorwise insertions.","feed_headline":"Maximal-contact counts diverge from local counts in the plane","feed_subtitle":"An extra 5-dimensional component makes naive and logarithmic counts differ on the plane with two lines as divisor.","key_machinery":"The central object is the moduli space $K^{\\max}_{0,2}(X|D,\\beta)$ of logarithmic stable maps with maximal contact at two marked points, together with its tropical cone complex $T_{0,2}(X,\\beta)$, the polyhedral parametrization of tropical curves by edge lengths and degrees. The load-bearing construction is the radial-alignment subdivision of $T_{0,2}(X,\\beta)$, which records a total order of the distances from the vertex carrying the first marking; combinatorially this subdivision is an iterated weighted stellar subdivision along floral cones, a floral cone being a cone indexed by a tropical curve type in which a contracted vertex carries the first marked point and is attached to several rational tails. Geometrically, the corresponding strata of the moduli space $K_{0,2}(X,\\beta)$ of stable maps are blown up in that order, so that the strict transforms of $K^{\\max}_{0,2}(X|D_1,\\beta)$ and $K^{\\max}_{0,2}(X|D_2,\\beta)$ meet transversely in the modified space. The classical blowup formula then expresses the discrepancy between this transverse intersection and the naive product as a sum of tautological correction terms, built from Chern classes of normal bundles, Segre classes of boundary strata, and descendent integrals.","core_discovery":"The paper's main negative result, Theorem X, states that the naive and logarithmic maximal-contact theories differ for $\\mathbb{P}^2$ with $D=H_1+H_2$ two lines: in degree $2$ the strong form of the local/logarithmic correspondence fails, and in degree $4$ the original form fails. The proof exhibits, in $K_{0,0}(\\mathbb{P}^2,4)$, that the image of the logarithmic moduli space is only one component of the intersection of the two images coming from the relative theories of $(\\mathbb{P}^2,H_1)$ and $(\\mathbb{P}^2,H_2)$; the intersection contains a second 5-dimensional component, consisting of maps with a contracted component mapping to $H_1\\cap H_2$ and four rational tails mapped isomorphically onto lines, and this component contributes with positive multiplicity. The conjectured class identity therefore cannot hold. In place of it, Theorem 3.6 gives a corrected product formula: after an explicit sequence of weighted blowups of $K_{0,2}(X,\\beta)$ along strict transforms of floral strata, the strict transforms of the two relative loci intersect transversely, and the difference between that transverse product and the original naive product is recorded by tautological correction terms. Theorem W establishes the numerical correspondence for pairs $(\\prod_i X_i,\\sum_i D_i)$ with primary factorwise insertions, giving the first non-toric instances in dimension greater than two.","pith_inferences":["The mechanism behind the counterexamples, the contracted component carrying a marking, suggests a threshold phenomenon: the naive and logarithmic classes should agree for any curve class where such floral configurations cannot occur. This is a testable extension, not a claim of the paper.","Because the counterexamples live in $\\mathbb{P}^2$ with toric boundary, a torus-localization computation of the degree-2 or degree-4 intersection would independently verify the dimension counts that the paper leaves to 'elementary geometry'; such a check is not performed in the paper.","The paper's observation that the product structure 'crushes' the corrections hints that the correspondence may hold for targets more general than literal products, namely those admitting a birational morphism that contracts the floral configurations; exploring this would be a natural continuation."],"forward_implications":["The two local/logarithmic correspondence conjectures are false in stated generality; multiplying the relative classes of individual divisor components does not in general produce the logarithmic maximal-contact class.","The strong form of the correspondence can fail earlier than the original form: in degree 2 for $\\mathbb{P}^2$ with two lines the strong form fails while the original form still holds there, and only degree 4 refutes the original form.","For section pairs, the difference between the logarithmic and naive/local maximal-contact invariants is algorithmically computable: it is a sum of tautological correction terms attached to floral strata, expressible through Chern and Segre classes and descendent integrals.","For products of smooth projective pairs $(\\prod_i X_i,\\sum_i D_i)$ with hyperplane sections, the numerical local/logarithmic correspondence holds with primary factorwise insertions, giving the first non-toric examples in dimension greater than two.","The method reduces genus-zero maximal-contact logarithmic questions for section pairs to Gromov–Witten theory of smooth pairs, providing an alternative to degeneration-based approaches."],"supporting_citations":[{"why":"Supplies the smooth-pair local/logarithmic correspondence used in Lemma 1.1 to identify the product of the two relative classes with the local class, and again in Lemma 3.1 to push the naive class to the local class.","marker":"[25]"},{"why":"Supplies the numerical balancing criterion that identifies the closure of the interior of the maximal-contact loci in the moduli space of stable maps, used in Lemma 1.3 to locate the second component of the intersection.","marker":"[12]"},{"why":"Supplies the intersection-theoretic tools used for the counterexample and the correction: the positive decomposition of an intersection into component classes in Proposition 1.2, and the blowup formula comparing strict and total transforms in Section 3.","marker":"[11]"},{"why":"Supplies weak semistable reduction, used to replace the morphism from the logarithmic moduli space to the moduli of stable maps by an integral and saturated birational model so that the corrected fibre product is Cartesian as an ordinary stack.","marker":"[3]"},{"why":"Supplies the canonical minimal weak semistable reduction, giving a unique subdivision and making the subsequent radial-alignment construction independent of auxiliary choices.","marker":"[16]"},{"why":"Supplies the moduli theory of logarithmic maps underlying the naive expectation that the relative loci intersect to give the logarithmic locus, and provides the fibre-product diagram of moduli spaces.","marker":"[1]"},{"why":"Supplies the product formula for Gromov–Witten invariants of product targets, used in Theorem 5.2 to identify the pushforward of the local class on a product with the product of local classes.","marker":"[7]"},{"why":"Supplies the virtual pullback formalism used in Section 4 to extend the corrected product formula from projective-space targets to general hyperplane sections by pulling back virtual classes.","marker":"[15]"}],"fun_headline_variants":["Counterexamples break local-log conjectures for maximal contacts","Maximal contact counts corrected by extra components","Logarithmic vs local: explicit difference via relative invariants","Plane counterexample reveals missing component in degree 4","Product formula fixes local-log discrepancy with blowups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The counterexamples rest on the assertion that no degenerate curve configuration contributes an overlap of dimension greater than 5 in the relevant intersection; the bound is argued by a case analysis and 'elementary geometry' rather than a systematic proof, and a single 6-dimensional boundary contribution would ruin the inequality.","fun_headline_variants_meta":{"raw":{"variants":["Counterexamples break local-log conjectures for maximal contacts","Maximal contact counts corrected by extra components","Logarithmic vs local: explicit difference via relative invariants","Plane counterexample reveals missing component in degree 4","Product formula fixes local-log discrepancy with blowups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000158,"raw_usage":{"total_tokens":1242,"prompt_tokens":981,"completion_tokens":261,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":185}},"tokens_in":597,"tokens_out":261,"duration_ms":3021,"temperature":1.0,"reasoning_tokens":185,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:38:45.778382+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the dimension of the boundary locus in $K_{0,0}(\\mathbb{P}^2,4)$ consisting of maps whose domain is a contracted component at $H_1\\cap H_2$ glued to four rational tails mapping isomorphically onto lines. Proposition 1.2 requires this locus to have dimension 5; an explicit parametrization (for instance by gluing parameters or equivariant localization) that produced dimension 6 would make the intersection improper and break the claimed inequality, while a clean dimension-5 computation would confirm the counterexample.","supporting_citations":[{"cited_title":"VAN GARREL , T","cited_arxiv_id":null,"evidence_quote":"Supplies the smooth-pair local/logarithmic correspondence used in Lemma 1.1 to identify the product of the two relative classes with the local class, and again in Lemma 3.1 to push the naive class to the local class."},{"cited_title":"G ATHMANN , Absolute and relative Gromov-Witten invariants of very ample hypersurfaces, Duke Math","cited_arxiv_id":null,"evidence_quote":"Supplies the numerical balancing criterion that identifies the closure of the interior of the maximal-contact loci in the moduli space of stable maps, used in Lemma 1.3 to locate the second component of the intersection."},{"cited_title":"F ULTON , Intersection theory, vol","cited_arxiv_id":null,"evidence_quote":"Supplies the intersection-theoretic tools used for the counterexample and the correction: the positive decomposition of an intersection into component classes in Proposition 1.2, and the blowup formula comparing strict and total transforms in Section 3."},{"cited_title":"A BRAMOVICH AND K","cited_arxiv_id":null,"evidence_quote":"Supplies weak semistable reduction, used to replace the morphism from the logarithmic moduli space to the moduli of stable maps by an integral and saturated birational model so that the corrected fibre product is Cartesian as an ordinary stack."},{"cited_title":"A BRAMOVICH AND Q","cited_arxiv_id":null,"evidence_quote":"Supplies the moduli theory of logarithmic maps underlying the naive expectation that the relative loci intersect to give the logarithmic locus, and provides the fibre-product diagram of moduli spaces."},{"cited_title":"B EHREND , The product formula for Gromov-Witten invariants, J","cited_arxiv_id":null,"evidence_quote":"Supplies the product formula for Gromov–Witten invariants of product targets, used in Theorem 5.2 to identify the pushforward of the local class on a product with the product of local classes."},{"cited_title":"M ANOLACHE , Virtual pull-backs, J","cited_arxiv_id":null,"evidence_quote":"Supplies the virtual pullback formalism used in Section 4 to extend the corrected product formula from projective-space targets to general hyperplane sections by pulling back virtual classes."}],"review_version":1}