{"id":"e819c636-2624-4b4f-86e4-1c49daafc278","arxiv_id":"1908.04371","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Q-factorial projective toric varieties whose toric boundary divisors are nef, the genus-zero log and local Gromov-Witten invariants with point and descendant insertions agree after the log-local normalization, and both are computed in closed form.","lead":"This mathematics paper proves that two different counting methods for rational curves on toric varieties, logarithmic and local Gromov-Witten theory, give the same answers after a predicted correction factor. It extends the log-local principle beyond smooth varieties and writes the counts in closed form for every degree.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Log-side proof relies on an unproved extension of [18] to singular fake weighted projective spaces; Remark 4.1's avoidance claim is false for P(1,1,2), d=1, and the zero-intersection vanishing uses a false virtual-dimension statement.","rationale":"The paper's central claim is the log-local relation in Theorem 3.3, with explicit closed forms for log invariants (Theorem 3.1) and local invariants (Theorem 3.2). The local side is derived from the mirror theorem for toric stacks [10] and is largely standard. The genuinely fragile input is the log side: all values Rp_d and Rq_d are obtained by applying [18, Theorem 1.1] and [19, Theorem 1.2] to products of fake weighted projective spaces. Since Proposition 2.1 shows the nef toric boundary condition forces exactly these singular targets, the smooth-variety statement of [18] cannot be assumed without proof. Remark 4.1 explicitly flags this and asserts an avoidance of deeper strata, but the assertion is not demonstrated and is in fact false in the simplest singular example P(1,1,2), d = 1, where every degree-one curve passes through the singular point. The separate claim in Section 5 that zero-intersection cases have negative virtual dimension is also internally inconsistent with the dimension formula of Section 2.2. These gaps do not show the formulas are wrong; the local-mirror computation and the final algebra in Theorem 3.3 are coherent. They do show that the proof as written is incomplete, which matches the reader's CONDITIONAL verdict. I would not move the verdict: the central construction is plausible and the gaps are local and repairable, but the manuscript should not be accepted without a written extension of [18] to the singular toric setting and a corrected vanishing argument.","tokens_in":17825,"tokens_out":25039,"duration_ms":295091,"concrete_test":"Compute the log invariant Rq_d for X = P(1,1,2), d = 1 directly from the log Gromov-Witten moduli space, for example by degenerating to the normal cone of the singular point [0:0:1] or by a direct evaluation of the obstruction theory, and compare with the value 2 predicted by Theorem 3.1. This example is decisive: every degree-one curve is a line ax + by = 0 and passes through the singular point, so the check tests both the extension of [18, Theorem 1.1] and the assertion of Remark 4.1 that curves avoid deeper strata.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.1 applies [18, Theorem 1.1] and [19, Theorem 1.2] to Q-factorial fake weighted projective spaces, which are singular for non-trivial weights or finite groups. Remark 4.1 concedes that [18] is stated for smooth varieties and asserts that curves never meet deeper toric strata, so the arguments carry through. This is the load-bearing step: all log values in Theorem 3.1 are obtained from it. The assertion is not obviously true; for X = P(1,1,2) in degree d = 1, every degree-one curve is a line ax + by = 0 and passes through the singular point [0:0:1], which is a deeper toric stratum (the intersection of two toric divisors). Thus the stated justification fails exactly in the singular cases the paper claims to cover. Moreover, the vanishing claim in Section 5 for d with some d·D_j = 0 says the virtual dimension is negative, contradicting the dimension formula vdim = n_X + m + l_D - 3 of Section 2.2, which is independent of d and positive. Both gaps affect the proof of Theorem 3.1 and therefore of Theorem 3.3.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an extension of the log-local principle of van Garrel--Graber--Ruddat to toric pairs (X,D) where X is a Q-factorial projective toric variety and D is the toric boundary. Under the assumption that every component of D is nef, Proposition 2.1 reduces X to a product of fake weighted projective spaces. The main result, Theorem 3.3, asserts that for every effective curve class d, the genus-zero maximally tangent log Gromov--Witten invariants with one or two point insertions and descendant insertions at one point satisfy the log-local relation N_d p_d = Rp_d and N_d q_d = Rq_d. The log side is computed in Theorem 3.1 using tropical correspondence and the multiplicity algorithm of [18,19], yielding Rp_d=1 and Rq_d equal to a product of group orders, weights, and d^{n_X}. The local side is computed in Theorem 3.2 via the mirror theorem for toric stacks, yielding closed hypergeometric formulas. The two sides are computed independently and then compared. The paper explicitly claims the correspondence holds for all degrees, including when some d.D_j=0, without invoking the original smoothness assumptions of Conjecture 1.1.","tokens_in":18086,"tokens_out":13673,"duration_ms":150165,"significance":"If the proof is completed, the paper gives a substantial generalization of the log-local principle: it replaces smooth X and normal-crossing D by possibly singular Q-factorial toric varieties with nef toric boundary, and it removes the positivity assumption d.D_j>0. The main formulas are explicit, closed-form, and parameter-free, which is a notable strength. The local computations are grounded in the established mirror theorem of Coates--Corti--Iritani--Tseng, and the log and local sides are genuinely computed by independent methods rather than fitted to each other. The paper also clearly situates itself relative to the parallel work of Nabijou--Ranganathan. However, the proof as written contains two load-bearing gaps: the extension of the tropical correspondence to singular targets is not justified, and the vanishing statement for d.D_j=0 is supported by a false dimension claim. These issues currently prevent the paper from fully establishing its central theorem.","major_comments":[{"comment":"The proof of the log-side formulas in Theorem 3.1 relies on applying [18, Theorem 1.1] and [19, Theorem 1.2] to Q-factorial fake weighted projective spaces, but [18] is stated for smooth varieties. Remark 4.1 asserts that in the cases of interest the curves never meet the deeper toric strata, so the arguments carry through. This assertion is false as stated: for X = P(1,1,2) and degree d = 1, a degree-one curve is given by an equation ax + by = 0, and every such curve passes through the singular point [0:0:1], which is the intersection of two toric divisors and hence a deeper toric stratum. Since all log invariants in Theorem 3.1 are obtained through this correspondence, the authors need either a proof of the correspondence for these singular toric varieties or a separate argument showing that the tropical counts compute the log invariants without the stated avoidance condition.","section":"Section 4.1, Remark 4.1"},{"comment":"The vanishing claim for d with some d.D_j = 0 is justified by saying that 'the virtual dimension of the moduli problem is negative in that case.' This contradicts the paper's own formula in Section 2.2, vdim = n_X + m + l_D - 3, which is independent of d and positive for the moduli spaces considered with m=1 or m=2. The invariant may indeed vanish, but the provided reason is internally inconsistent. A correct argument is needed, for instance by analyzing the contact-order zero conditions or by a deformation argument, because this vanishing is part of the statement of Theorem 3.1 and is used in Theorem 3.3 for the case d.D_j=0.","section":"Section 5, first paragraph"},{"comment":"The local invariants p_d and q_d are defined in Section 2.3 only under the hypothesis d.D_j > 0 for all j, since R^1 pi_* f^* O(-D_j) is then a vector bundle and the virtual class (2.7) is defined via its top Chern class. However, Theorem 3.2 states formulas for all effective curve classes d, including those with d.D_j = 0, and Theorem 3.3 explicitly claims the log-local relation with no assumptions on d.D_j. The manuscript does not define the local invariants in the case d.D_j = 0, nor does it explain how the hypergeometric formulas in Theorem 3.2 are to be interpreted geometrically in that case. This is a load-bearing point for the 'no assumptions on d.D_j' part of the main theorem and must be addressed.","section":"Sections 2.3 and 3.2"}],"minor_comments":[{"comment":"There are small typos: 'hold s' in the abstract and 'assumtions' in Theorem 3.3 should be 'holds' and 'assumptions'.","section":"Abstract and Theorem 3.3"},{"comment":"The notation for the monomial product after defining x = alpha_{j_1}^{1} ... is confusing: the displayed expression writes x^n but the variable v in N^m is not used consistently. Clarifying the notation would improve readability.","section":"Section 2.1"},{"comment":"The tropical curve uniqueness arguments are terse; for instance, Proposition 5.1 asserts the unique element without discussing why the fixed general point condition is compatible with the star-shaped tropical curve. Adding a sentence explaining the general position argument would help.","section":"Section 5, Propositions 5.1-5.4"},{"comment":"Remark 6.5 leaves the computation of two-point descendant invariants with distributed psi-powers to the reader. Since the paper's main theorem is about descendent insertions at one point only, this is acceptable, but a brief statement of the expected result would make the scope clearer.","section":"Section 6, Remark 6.5"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a promising and likely correct result, but the proof as written has two significant gaps: the singular extension of the tropical correspondence is not justified (and the Remark 4.1 justification is false), and the vanishing for d.D_j=0 is supported by a self-contradictory dimension statement. The local-side formulas may also require a clarification of the meaning of local invariants when some d.D_j=0. These issues are fixable in principle, but they require substantive additional work rather than minor editing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real extension of the log-local principle to Q-factorial toric pairs, with explicit all-degree closed forms for both the log and local invariants. The local computation via the CCIT mirror theorem looks solid and is independent of the log side. The log side, however, is the weak link. Theorem 3.1's proof leans on [18] for singular toric varieties, and Remark 4.1 tries to justify this by asserting the relevant curves never meet the deeper toric strata. That assertion is false: for X=P(1,1,2), d=1, every line ax+by=0 passes through the singular point [0:0:1], which is the intersection of two boundary divisors. So the tropical correspondence cannot be applied by the argument given. The vanishing statement for d·D_j=0 is also wrong as written: the virtual dimension is n_X+m+l_D-3, which is independent of d and positive, not negative. And the local invariants p_d, q_d are only defined in Section 2.3 under d·D_j>0, yet Theorem 3.2 states formulas for all effective d. Those are three genuine gaps in the proof of the main theorems. None of this looks like fitting or circularity: both sides are computed from independent tools, and the closed forms are concrete and checkable. The local mirror computation and the multiplicities in the tropical count are worked out in detail. But the log-side derivation as written does not cover the singular cases the paper advertises. A revision that either proves the tropical correspondence for these toric pairs or states Theorem 3.1 for the cases where it holds, and that fixes the d·D_j=0 discussion, could turn this into a solid paper. As it stands, I'd send it to a competent referee rather than desk reject—the conjectures and formulas are important enough that a referee should sort out whether the gaps are repairable. I would not cite the log-side proof in its current form.","headline":"A serious extension of the log-local principle with closed forms for all degrees, but the log-side proof as written has a false avoidance claim and a wrong vanishing argument that need repair.","tokens_in":18585,"tokens_out":11349,"would_cite":false,"duration_ms":119517,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14M25","14T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For toric varieties with nef boundary, log and local Gromov-Witten counts match for every effective degree.","keywords":["log Gromov-Witten invariants","local Gromov-Witten invariants","toric boundary","maximal tangency","fake weighted projective spaces","tropical correspondence principle","mirror symmetry for toric stacks","nef toric pairs"],"falsifier":"Take $X=\\mathbb{P}(1,1,2)$ and degree $d=1$. The theorem predicts exactly one maximally tangent tropical curve through two general points, of multiplicity $2$, so the two-point log invariant equals $2$. An independent computation, for instance by degeneration to the toric boundary or by direct tropical enumeration that allows contact with the $\\mathbb{Z}/2$ orbifold point, that yields any other value would falsify the claimed extension to singular toric pairs.","tokens_in":17620,"feed_emoji":"📐","tokens_out":14615,"duration_ms":142787,"temperature":0.7,"pith_summary":"The paper establishes the log-local principle for toric pairs $(X,D)$ in which $X$ is a possibly singular $\\mathbb{Q}$-factorial projective toric variety and every component of the toric boundary $D$ is nef, meaning numerically nonnegative on curves. Such pairs are forced to be products of fake weighted projective spaces, and the paper computes both sides of the correspondence in closed form: the genus-zero log Gromov-Witten invariants of maximal tangency (each boundary divisor met at one point with maximal contact) and the genus-zero local invariants of the total space of $\\bigoplus_j \\mathcal{O}(-D_j)$. The result is that, after multiplying the local invariants by the explicit factor $N_X^d=\\prod_j (-1)^{d\\cdot D_j+1}d\\cdot D_j$, the two sides agree for every effective curve class $d$, with up to two point insertions and one descendant $\\psi$-insertion. This extends the known smooth normal-crossing case to singularities and to boundaries that are not normal crossing.","feed_headline":"Log and local curve counts agree on toric boundaries","feed_subtitle":"Maximal-tangency log invariants equal twisted local invariants up to a universal factor for every degree.","key_machinery":"The paper's computations are carried by the tropical correspondence principle and the equivariant mirror theorem for toric stacks. Tropical correspondence identifies the log invariants with weighted counts of maximally tangent rational tropical curves, and the multiplicity algorithm computes each curve's weight by iterated exterior contractions in the dual lattice. The mirror theorem identifies the small $J$-function of the local geometry with the stacky $I$-function, a hypergeometric series whose coefficients are products over the intersection numbers $e^X_j(d)$; comparing its $z$-expansion isolates the descendant invariants. The link between the two sides is the normalization factor $N_X^d$, and the classification of nef toric pairs as products of fake weighted projective spaces (quotients of weighted projective spaces by finite abelian groups) reduces every computation to the weights $(w^{(i)})_j$ and group orders $|G_i|$.","core_discovery":"The central claim is Theorem 3.3: for every nef toric pair and every effective curve class $d$, the identities $N_X^d\\, p^X_d = Rp^X_d$ and $N_X^d\\, q^X_d = Rq^X_d$ hold. The paper evaluates both sides. The one-point log invariant $Rp^X_d$ is $1$; the two-point log invariant $Rq^X_d$ is $\\prod_i |G_i|\\prod_{i,j}(w^{(i)})_j\\, d^{n_X}$. The local invariants are $p^X_d = (-1)^{e^X(d)-n_X-r_X}/\\prod_j^\\circ e^X_j(d)$ and $q^X_d = \\prod_i |G_i|\\prod_{i,j}(w^{(i)})_j\\, d^{n_X} p^X_d$, where $e^X_j(d)=d\\cdot D_j$ and $e^X(d)=-d\\cdot K_X$. The theorem needs no positivity assumption on $d\\cdot D_j$: when some intersection number vanishes, both log and local invariants vanish and the identities still hold.","pith_inferences":["The paper proves the two-point identity only with the descendant $\\psi^{r_X-1}$ placed at one marked point; the same closed-form structure suggests the equality persists for arbitrary distributions of $\\psi$-classes among the two points, which the paper itself leaves as an exercise from the symplectic S-matrix.","If the extended tropical correspondence is valid, the log invariants are determined entirely by the intersection numbers $e^X_j(d)=d\\cdot D_j$; one could test this by enumerating tropical curves for a toric surface with the same boundary intersection pattern but a different fan.","The rational closed forms for the local invariants give concrete base cases for the refined/BPS formulation of the log-local principle, since expanding them in the degree produces the integrality statements one would expect from the log side."],"forward_implications":["For every effective curve class $d$ on a nef toric pair, the genus-zero log invariant with one maximal-tangency point and $\\psi^{n_X+r_X-2}$ is exactly $1$.","The two-point log invariant equals $(\\prod_i |G_i|)(\\prod_{i,j} (w^{(i)})_j)\\, d^{n_X}$, so it depends only on weights, group orders, and the degree, not on the finer structure of the fan.","The local invariants are given in closed rational form by $p^X_d = (-1)^{e^X(d)-n_X-r_X}/\\prod_j^\\circ e^X_j(d)$, and the universal multiplicative relation with the log invariants holds in all degrees.","The correspondence holds without assuming $d\\cdot D_j>0$: when some intersection number vanishes, the virtual dimension drops and both log and local invariants vanish separately.","This extends the smooth normal-crossing case of the log-local principle to singular $\\mathbb{Q}$-factorial toric varieties with non-normal-crossing toric boundary."],"supporting_citations":[{"why":"It formulates the log-local conjecture and proves it for smooth pairs, the statement this paper extends.","marker":"[12]"},{"why":"It supplies the tropical correspondence principle that converts the log Gromov-Witten invariants into weighted counts of tropical curves.","marker":"[18]"},{"why":"It gives the tropical multiplicity algorithm used to evaluate the unique tropical curve counts.","marker":"[19]"},{"why":"It provides the equivariant mirror theorem identifying the small J-function with the stacky I-function, used to compute the local invariants.","marker":"[10]"},{"why":"It is the classification result used to conclude that a toric variety whose boundary components are nef is a product of fake weighted projective spaces.","marker":"[11]"},{"why":"It defines the orbifold Gromov-Witten theory used to give meaning to the local geometry when X is singular.","marker":"[2]"},{"why":"It establishes the orbifold Gromov-Witten invariants and Chen-Ruan pairing underlying the local invariants.","marker":"[6]"},{"why":"It supplies the cohomology formula identifying the point class with $\\prod_i |G_i|\\prod_{i,j}(w^{(i)})_j H^{n_X}$.","marker":"[16]"}],"fun_headline_variants":["Toric log-local principle proven for all nef pairs","Log and local GW invariants coincide on toric boundary","Maximal-tangency log = twisted local at every degree","Extension of log-local principle holds for toric boundary"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that a tropical curve-counting correspondence stated for smooth varieties remains valid for the singular fake weighted projective spaces that occur here, because the relevant curves avoid the deeper toric strata; the paper flags this in Remark 4.1 but does not prove it.","fun_headline_variants_meta":{"raw":{"variants":["Toric log-local principle proven for all nef pairs","Log and local GW invariants coincide on toric boundary","Maximal-tangency log = twisted local at every degree","Extension of log-local principle holds for toric boundary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001469,"raw_usage":{"total_tokens":5896,"prompt_tokens":926,"completion_tokens":4970,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":4903}},"tokens_in":542,"tokens_out":4970,"duration_ms":32259,"temperature":1.0,"reasoning_tokens":4903,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:45:32.369315+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $X=\\mathbb{P}(1,1,2)$ and degree $d=1$. The theorem predicts exactly one maximally tangent tropical curve through two general points, of multiplicity $2$, so the two-point log invariant equals $2$. An independent computation, for instance by degeneration to the toric boundary or by direct tropical enumeration that allows contact with the $\\mathbb{Z}/2$ orbifold point, that yields any other value would falsify the claimed extension to singular toric pairs.","supporting_citations":[{"cited_title":"van Garrel, T","cited_arxiv_id":null,"evidence_quote":"It formulates the log-local conjecture and proves it for smooth pairs, the statement this paper extends."},{"cited_title":"Coates, A","cited_arxiv_id":null,"evidence_quote":"It provides the equivariant mirror theorem identifying the small J-function with the stacky I-function, used to compute the local invariants."},{"cited_title":"Fujino and H","cited_arxiv_id":null,"evidence_quote":"It is the classification result used to conclude that a toric variety whose boundary components are nef is a product of fake weighted projective spaces."},{"cited_title":"Abramovich, T","cited_arxiv_id":null,"evidence_quote":"It defines the orbifold Gromov-Witten theory used to give meaning to the local geometry when X is singular."},{"cited_title":"Chen and Y","cited_arxiv_id":null,"evidence_quote":"It establishes the orbifold Gromov-Witten invariants and Chen-Ruan pairing underlying the local invariants."},{"cited_title":"Kaw asaki, Cohomology of twisted projective spaces and lens complexe s, Math","cited_arxiv_id":null,"evidence_quote":"It supplies the cohomology formula identifying the point class with $\\prod_i |G_i|\\prod_{i,j}(w^{(i)})_j H^{n_X}$."}],"review_version":1}