{"id":"f02915e4-680e-437e-b98f-cb863ab4af5e","arxiv_id":"1908.10906","paper_version":5,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves explicit multiplicity formulas for non-rigid A1-curves and for unions of two rigid A1-curves in maximal-tangency genus 0 log Gromov-Witten invariants on surfaces.","lead":"This paper computes the contributions of two new classes of curves to genus 0 log Gromov-Witten invariants on surfaces, one involving a logarithmic moduli space of one-dimensional sheaves and the other a full deformation calculation for unions of two rigid curves. The results turn previously abstract invariants into explicit numbers and expose differences between log and relative stable map counts.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.14's count depends on the explicit genericity assumption (Z1.Z2)_P = min(d1,d2), which Remark 1.15 only conjectures for general D; the proof and its applications are conditional on an unproved hypothesis.","rationale":"The theorem is carefully proved under the explicit hypothesis (Z1.Z2)_P = min(d1,d2), and the proof uses that hypothesis precisely where needed: Lemma 5.16 converts it into the coefficient inequality that forces v = 0, thereby preventing an obstructed extension and fixing the length of the local moduli component at min{e1,e2}. I found no hidden gap in the deformation-theoretic computation: the coordinates in Notation 5.9, the conditions in Lemma 5.11, and the extension analysis in Section 5.5 are internally consistent, and the count of d isomorphism classes of log maps matches the automorphism calculation in Theorem 5.12(3). The reader's weakest-assumption diagnosis is accurate: the condition (Z1.Z2)_P = min(d1,d2) is the single most load-bearing premise, and it is not proven to hold for general D; Remark 1.15 explicitly flags it as an expectation rather than a theorem. This does not invalidate Theorem 1.14 as a conditional statement, but it means any application to a concrete pair (X,D) must verify the genericity condition, and the paper's broader enumerative claims for 'general D' rest on an unproved genericity assertion. The reader's ACCEPT verdict with moderate confidence is therefore appropriate, and no adjustment is needed.","tokens_in":42261,"tokens_out":24656,"duration_ms":277961,"concrete_test":"For a concrete one-parameter family of pairs (X, D_t) with D_t anticanonical and two A1-curves Z1(t), Z2(t) meeting D_t at P with d1 = d2 = d, compute local analytic equations y = a_i(t) x^d + ... and determine the locus where a_1(t) = a_2(t). If this higher-contact locus has positive codimension in the parameter t, then the expectation of Remark 1.15 is confirmed for that family; if it is dense or codimension 0, the genericity condition fails and the theorem's enumerative conclusion cannot be applied without additional hypotheses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Lemma 5.16 in Section 5.5: to rule out an extension from S_{e1-1} to S_{e1}, the proof needs v = 0, and for e1 = e2 = 1 this follows from the inequality A1(0)(B'_2(0))^d != A2(0)(B'_1(0))^d, which is exactly the statement that (Z1.Z2)_P = d. If the two curves have higher-order contact at P, this inequality can fail, v need not vanish, and the deformation-theoretic conclusion that f_{min{e1,e2}-1,p} is an isomorphism onto a connected component is no longer established. The claimed number d of log maps and their multiplicity min{e1,e2}, hence the contribution min{d1,d2} to N_{beta1+beta2}(X,D), therefore depend on this genericity condition. The theorem is internally sound under its stated assumption, but Remark 1.15 only 'expects' the condition for general D and gives no proof; in applications one must verify the condition for the specific curves in question.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies genus 0 maximally tangent stable log maps to a smooth surface with a divisor D, focusing on the contributions of zero-dimensional components of the moduli space to the log Gromov-Witten invariants N_beta(X,D). It proves two main results. (A) Corollary 1.12 states that an irreducible rational A1-curve C that is smooth at the contact point P contributes its ordinary stable map multiplicity l(C) to N_beta; this is deduced from the smoothness of a newly introduced moduli space MMI_beta of 'sheaves of maximal intersection' (Theorem 3.12). (B) Theorem 1.14 states that for two immersed rigid A1-curves Z1, Z2 satisfying (K_X+D).Z_i=0 and meeting D at the same smooth point P with intersection multiplicity (Z1.Z2)_P = min(d1,d2), there are exactly d=gcd(d1,d2) basic stable log maps with image Z1 union Z2, each isolated with multiplicity min(e1,e2), so together they contribute min(d1,d2) to the invariant. The proof occupies Section 5 and is an explicit coordinate computation of the log structures and deformation theory. Section 6 compares the result with relative stable maps in a concrete example.","tokens_in":42509,"tokens_out":11642,"duration_ms":113769,"significance":"If correct, these are the first explicit computations of log Gromov-Witten contributions for non-rigid and reducible image curves. The introduction of the moduli space MMI_beta and its nonsingularity (Theorem 3.12) is a novel sheaf-theoretic tool that parallels the K3 surface story. The results feed directly into applications to log BPS numbers and local BPS invariants, as illustrated in Section 2.3, and the paper includes a fully worked example recovering N_beta=18 for a weak del Pezzo surface, with a comparison to tropical multiplicity. The proofs are lengthy, detailed, and transparent, using explicit coordinate models for the log structures, and the paper contains a useful example contrasting log and relative stable maps (Section 6).","major_comments":[{"comment":"The proof that v=0 in the case n=e1 uses the inequality A1(0)(B'_2(0))^d != A2(0)(B'_1(0))^d, which is equivalent to the genericity hypothesis (Z1.Z2)_P = min(d1,d2). For d1 != d2 this condition is automatic (Remark 1.15), but for d1 = d2 it is an extra assumption that is only conjectured to hold for general D. The applications in Section 2.3 and Example 6.1 satisfy the condition, but the paper should explicitly state that Theorem 1.14 is conditional on this hypothesis and that it must be verified in each equal-degree application. This is a limitation of the theorem's scope rather than an error in the proof.","section":"Section 5.5, Lemma 5.16"}],"minor_comments":[{"comment":"The phrase 'remedy to that shortcoming' should be 'remedy that shortcoming'.","section":"Introduction, Section 1.1"},{"comment":"The computation of dim Ext^1(F,F(-D)) = beta^2 + 1 would be clearer with an explicit expansion of the Riemann-Roch calculation; the current sentence is terse.","section":"Section 3, proof of Theorem 3.12"},{"comment":"For N=2, the notation Bji is later abbreviated to Bi; a brief remark that B_i = B_2i in the rest of Section 5 would improve readability.","section":"Section 5, Notation 5.9"},{"comment":"It would be helpful to explicitly state that the two choices u2 = +/-1/2 correspond to the two log maps counted by d=2 and that (Z'_1.Z'_2)_P = 2 verifies the genericity hypothesis of Theorem 1.14.","section":"Section 6, Example 6.1"}],"recommendation":"accept","confidential_remarks":"This is a strong paper with rigorous, explicit proofs. The genericity condition in Theorem 1.14 is clearly stated and its role is correctly identified; the applications either satisfy it automatically or verify it explicitly. I support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. This is a serious paper that actually computes log Gromov-Witten multiplicities, not just formal structure. And its main two-curve formula is proved under an explicit genericity assumption that the authors conjecture but do not establish for general D. That is the caveat.\n\nThe genuinely new object is the moduli space MMI_beta of sheaves of maximal intersection. The nonsingularity theorem (1.11/3.12) is a real result, and it yields Corollary 1.12: for an integral rational A1-curve smooth at P, the contribution to the maximal-tangency log GW invariant is just its classical stable map multiplicity l(C), with no new log phenomena. That is clean and useful, and it mirrors the K3 story nicely. The second half gives an explicit description of stable log maps with image Z1 ∪ Z2: d = gcd(d1,d2) log maps, each with length min(e1,e2), so the total contribution is min(d1,d2). The comparison with relative stable maps—one point of length min(d1,d2) versus d points—is instructive and the worked example in Section 6 makes it concrete.\n\nThe proof is long and technical; I did not machine-check every Ext computation or formal series identity, but the structure is coherent and the intermediate lemmas are explicit. The soft spot is real: in the equal-degree case, Lemma 5.16 relies on the inequality A1(0)(B'_2(0))^d ≠ A2(0)(B'_1(0))^d to get v=0 and fix the multiplicity. If Z1 and Z2 have higher-order contact at P, that inequality can fail, and the count could change. Remark 1.15 'expects' the condition for general D, but there is no proof. So Theorem 1.14 is sound under its stated hypothesis; it is just not a fully unconditional statement, and applications must verify the hypothesis or avoid the equal-degree case. Also worth noting: some enumerative applications rely on Conjecture 1.8, but that is flagged and is already a theorem for P^2.\n\nThe citation pattern is fine—self-citations point to published background results. This paper deserves a serious referee. I would send it out, with the main request being to sharpen Remark 1.15, prove the genericity condition in the needed cases, or at least state clearly in the applications where it is being assumed.","headline":"Real new results on log GW multiplicities; main caveat is an unproved genericity assumption in the two-curve formula.","tokens_in":43021,"tokens_out":4180,"would_cite":true,"duration_ms":40443,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14A21","14B10","14D15","14D20","14J26"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two glued rational curves contribute exactly the smaller of their two tangency orders to the log Gromov-Witten count, with each of the d log maps carrying multiplicity min(e1,e2).","keywords":["log Gromov-Witten invariants","stable log maps","maximal tangency","log Calabi-Yau surfaces","sheaves of maximal intersection","multiplicities","relative stable maps","log BPS numbers"],"falsifier":"Compute the same local contribution in a one-parameter family of divisors D in which two A1-curves develop contact at P of order greater than min(d1,d2), using the deformation equations of Section 5; Theorem 1.14 predicts exactly d isolated log maps of length min(e1,e2), so any other count would show that the generic-intersection hypothesis is doing real work.","tokens_in":42087,"feed_emoji":"🧮","tokens_out":11063,"duration_ms":107408,"temperature":0.7,"pith_summary":"This paper computes the numerical contributions of two natural kinds of zero-dimensional components in the moduli space of genus-0 stable log maps of maximal tangency on a surface. It establishes that an irreducible $A^1$-curve contributes to the log Gromov-Witten invariant exactly the same multiplicity it has as an ordinary stable map, namely the length $l(C)$ of the moduli space of maps to the curve. It then establishes that when two distinct immersed rational $A^1$-curves $Z_1,Z_2$ are maximally tangent to the boundary divisor at the same smooth point and meet there generically, their union supports exactly $\\gcd(d_1,d_2)$ stable log maps, each of multiplicity $\\min(e_1,e_2)$, for a total contribution of $\\min(d_1,d_2)$. These calculations matter because log Gromov-Witten invariants have many structural theorems but very few worked-out computations; the results supply explicit terms in decompositions of log BPS numbers and connect the log, relative, and tropical counting formalisms.","feed_headline":"Two tangent curves contribute min(d1,d2) to log counts","feed_subtitle":"New sheaf moduli and explicit log deformation theory fix the multiplicity of two maximal-tangency curves glued at a point.","key_machinery":"The load-bearing object in the first half is the moduli space $\\mathrm{MMI}_\\beta$ of sheaves of maximal intersection: torsion-free rank-one sheaves $F$ supported on integral curves of class $\\beta$ whose restriction to $D$ is $\\mathcal{O}_{wP}$ with $w=\\beta\\cdot D$. Theorem 1.11 proves that $\\mathrm{MMI}_\\beta$ and its point-fixed variant are nonsingular of dimension $2p_a(\\beta)=\\beta^2-w+2$, a logarithmic analogue of unobstructedness for sheaves on K3 surfaces; this smoothness is what forces infinitesimal deformations of a log map with image $C$ to factor through $C$, yielding Corollary 1.12. The second half carries out a fully explicit log deformation theory. The relevant domain is a chain $C_1\\cup C_0\\cup C_2$ with the middle component collapsed to $P$, the basic log structure is encoded by the coprime pair $(e_1,e_2)$ and by a $d$-th root $c_p$, and the extension calculation in Section 5 shows exactly which lifts survive to order $\\min(e_1,e_2)$, giving $d$ isolated points of length $\\min(e_1,e_2)$.","core_discovery":"On a smooth surface $X$ with an effective divisor $D$, the paper studies genus-0 basic stable log maps of maximal tangency to $D$. Corollary 1.12 states that if $X$ is a smooth projective rational surface, $K_X+D\\sim 0$, $P\\in D_{\\mathrm{sm}}$, and $C$ is an irreducible rational curve of class $\\beta$ maximally tangent to $D$ at $P$ and smooth at $P$, then the normalization map $\\mathbb{P}^1\\to C$ contributes $l(C)$ to $N_\\beta(X,D)$, where $l(C)$ is the Euler characteristic of the compactified Jacobian of $C$. Theorem 1.14 concerns two proper integral rational curves $Z_1,Z_2$ with $(K_X+D)\\cdot Z_i=0$, both maximally tangent to $D$ at the same point $P$, with immersive normalizations and with $(Z_1.Z_2)_P=\\min(d_1,d_2)$ for $d_i=D\\cdot Z_i$. Writing $d_1=de_1$, $d_2=de_2$ with $\\gcd(e_1,e_2)=1$, the theorem asserts that the stack $M_{\\beta_1+\\beta_2}(X,D)$ contains exactly $d$ stable log maps with image $Z_1\\cup Z_2$, each an isolated point of length $\\min(e_1,e_2)$; their combined contribution to $N_{\\beta_1+\\beta_2}(X,D)$ is therefore $\\min(d_1,d_2)$. The paper also shows by example that the relevant component of the log moduli space is not isomorphic to the corresponding relative stable map component, even though the two theories give the same numerical invariants.","pith_inferences":["If the smoothness of $\\mathrm{MMI}_\\beta$ is a genuine logarithmic analogue of K3 sheaf unobstructedness, then a sheaf-theoretic Euler-characteristic count should reproduce the log BPS numbers after the same kind of substitution used in the local BPS correspondence; the paper does not develop this.","The separation of the total contribution into $d$ log lifts times a length $\\min(e_1,e_2)$ suggests that for a union of $k$ maximally tangent components the count may be governed by the gcds of the tangency orders; the three-line case from Section 2.2.4, whose contribution is 3, is a concrete place to test this.","If Condition ($\\bullet$) fails, the clean sum in Corollary 1.13 could acquire correction terms from curves with worse singularities at $P$; a low-degree exhaustive enumeration on a del Pezzo surface would show whether such corrections actually occur.","The root-of-unity data $c_p$ distinguishing the $d$ log maps have no relative counterpart, so refined invariants that keep track of these data might be visible tropically as a choice of edge weights in a scattering diagram."],"forward_implications":["Under Condition ($\\bullet$), Corollary 1.13 gives an enumerative meaning to the log BPS number at a $\\beta$-primitive point: it is the sum of $l(C)$ over rational unibranch curves in the linear system, subject to Conjecture 1.8.","In the tropical-vertex example obtained by blowing up $\\mathbb{P}^2$ at six points, combining Corollary 1.12 with Theorem 1.14 reproduces $N_\\beta(S,D)=18$, matching the tropical multiplicity count and confirming deformation invariance through Euler numbers.","The log and relative moduli spaces are locally non-isomorphic near a two-component image: there are $d$ log maps of length $\\min(e_1,e_2)$ versus one relative map of length $\\min(d_1,d_2)$, yet by the comparison theorem the total contribution to the invariant agrees.","At a flex point of $(\\mathbb{P}^2,E)$, the decomposition takes the form $113 = 5 + 24 + 2\\cdot \\mathrm{Contr} + k_5$, so determining the remaining component would determine $k_5$, the number of degree-5 rational curves maximally tangent at a flex."],"supporting_citations":[{"why":"Defines basic stable log maps and their virtual fundamental class, the objects whose contributions are computed.","marker":"[28]"},{"why":"Establishes the stable log map moduli theory for Deligne-Faltings pairs used for the divisorial log structure.","marker":"[17]"},{"why":"Extends the moduli theory to Deligne-Faltings pairs II and supplies properness tools for maximal tangency.","marker":"[1]"},{"why":"The authors' previous paper gives the log BPS framework, classifies domain components, and supplies Proposition 1.7(3), from which Corollary 1.12 is drawn.","marker":"[21]"},{"why":"Computes multiple-cover contributions over rigid curves in the tropical vertex setting that the present paper extends to non-rigid and two-component images.","marker":"[27]"},{"why":"Identifies the multiplicity l(C) of a rational curve with the Euler characteristic of its compactified Jacobian, the number appearing in Corollary 1.12.","marker":"[22]"},{"why":"Provides the theorem on the relative compactified Picard scheme used to compute the dimension and nonsingularity of MMI_beta in Theorem 1.11.","marker":"[4]"},{"why":"Proves that the forgetful morphism from stable log maps to stable maps is finite, allowing zero-dimensional log components to be analyzed as lengths at points.","marker":"[60]"},{"why":"Comparison theorem guaranteeing that log and relative Gromov-Witten invariants agree, so the local difference exhibited in Example 6.2 does not change totals.","marker":"[3]"},{"why":"Gives the analogous multiplicity for reducible relative stable maps, one map of length min(d1,d2), which contrasts with the d log maps of length min(e1,e2).","marker":"[54]"}],"fun_headline_variants":["Single irreducible curve contributes l(C) to log counts","Two tangent curves sum to min(d1,d2) in log counts","Sheaves of maximal intersection compute log multiplicities","Log and relative stable maps differ but counts agree","Explicit log deformation theory fixes curve contributions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The count rests on the assertion that the two curves meet each other at P in the least special way possible, namely with intersection multiplicity equal to the smaller of their two tangency orders, which the authors conjecture holds for a general divisor D but do not prove.","fun_headline_variants_meta":{"raw":{"variants":["Single irreducible curve contributes l(C) to log counts","Two tangent curves sum to min(d1,d2) in log counts","Sheaves of maximal intersection compute log multiplicities","Log and relative stable maps differ but counts agree","Explicit log deformation theory fixes curve contributions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000759,"raw_usage":{"total_tokens":3450,"prompt_tokens":1101,"completion_tokens":2349,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":717,"completion_tokens_details":{"reasoning_tokens":2273}},"tokens_in":717,"tokens_out":2349,"duration_ms":19172,"temperature":1.0,"reasoning_tokens":2273,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:30:12.102483+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same local contribution in a one-parameter family of divisors D in which two A1-curves develop contact at P of order greater than min(d1,d2), using the deformation equations of Section 5; Theorem 1.14 predicts exactly d isolated log maps of length min(e1,e2), so any other count would show that the generic-intersection hypothesis is doing real work.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines basic stable log maps and their virtual fundamental class, the objects whose contributions are computed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the stable log map moduli theory for Deligne-Faltings pairs used for the divisorial log structure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the moduli theory to Deligne-Faltings pairs II and supplies properness tools for maximal tangency."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The authors' previous paper gives the log BPS framework, classifies domain components, and supplies Proposition 1.7(3), from which Corollary 1.12 is drawn."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes multiple-cover contributions over rigid curves in the tropical vertex setting that the present paper extends to non-rigid and two-component images."},{"cited_title":"Algebraic Geom","cited_arxiv_id":null,"evidence_quote":"Identifies the multiplicity l(C) of a rational curve with the Euler characteristic of its compactified Jacobian, the number appearing in Corollary 1.12."},{"cited_title":"Real and complex singularities (Proc. Ninth Nordic Summer School/NA VF Sympos. Math., Oslo, 1976)","cited_arxiv_id":null,"evidence_quote":"Provides the theorem on the relative compactified Picard scheme used to compute the dimension and nonsingularity of MMI_beta in Theorem 1.11."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that the forgetful morphism from stable log maps to stable maps is finite, allowing zero-dimensional log components to be analyzed as lengths at points."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Comparison theorem guaranteeing that log and relative Gromov-Witten invariants agree, so the local difference exhibited in Example 6.2 does not change totals."},{"cited_title":"On the multiplicity of reducible relative stable morphisms","cited_arxiv_id":"1711.08173","evidence_quote":"Gives the analogous multiplicity for reducible relative stable maps, one map of length min(d1,d2), which contrasts with the d log maps of length min(e1,e2)."}],"review_version":1}