{"id":"668a56e1-9a4a-48b4-bdaa-3f7e18afefe7","arxiv_id":"2505.14672","paper_version":4,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces complete quasimaps to Bl_{P^s}(P^r) via blow-up of Ciocan-Fontanine--Kim quasimaps and conjectures their tautological intersections enumerate curves of fixed complex structure under incidence conditions.","lead":"The paper constructs a new moduli space of complete quasimaps to the blow-up of projective space by blowing up an existing quasimap space at points where sections become linearly dependent. A smart generalist might read it to see how algebraic geometers are refining tools for counting curves with fixed shape on varieties.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Applicability of Farkas' purity result after blowing up the quasimap space is unverified","rationale":"The reader's weakest assumption correctly flags purity as load-bearing. The sharper issue is whether Farkas' theorem applies verbatim after the blow-up step that defines the new space; the paper states the guarantee without recording the required hypothesis check. This is internal to the argument rather than a disagreement with external consensus. The low-dimensional proof does not automatically extend, so the conjecture remains conditional on this verification. Verdict therefore moves from UNVERDICTED to CONDITIONAL.","tokens_in":1651,"tokens_out":403,"duration_ms":40212,"concrete_test":"For r=3, s=1, d=2 and target dimension 3, compute the actual dimension of the blown-up space at a general point of the linear-dependence locus and compare with the expected dimension formula stated in the paper; if the computed dimension differs by more than 0, the purity claim fails for the blown-up space.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central construction blows up Ciocan-Fontanine--Kim quasimaps at linear-dependence loci to produce complete quasimaps to Bl_{P^s}(P^r). The conjecture equates tautological intersections on this space with enumerative counts of fixed-complex-structure curves on X under general incidences. This requires the moduli spaces to be pure of expected dimension so that ordinary intersection theory (rather than virtual classes) applies. The paper invokes a result of Farkas for this purity but does not record an explicit check that the blow-up centers have the expected codimension or that the resulting space satisfies the hypotheses of Farkas' theorem (e.g., irreducibility, mild singularities, or the precise form of the quasimap data). If the blow-up alters dimension or introduces excess components outside the scope of the cited result, the intersection numbers cease to be enumerative in the claimed sense.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces a moduli space of complete quasimaps to Bl_{P^s}(P^r) obtained by blowing up the Ciocan-Fontanine--Kim quasimap space at loci of linear dependence among sections. It conjectures that tautological intersection numbers on this space enumerate curves of fixed complex structure on the target subject to general incidence conditions. Purity of expected dimension is asserted via a result of Farkas, and the conjecture is proved in dimension 2 using a Brill-Noether theorem for general curves on toric surfaces.","tokens_in":1844,"tokens_out":418,"duration_ms":44625,"significance":"If the conjecture holds, the construction supplies a new compactification suited to enumerative problems with fixed complex structure, offering a contrast to traditional moduli of maps. The explicit proof in dimension 2 and the appeal to Farkas' purity constitute concrete strengths that ground the approach in existing results.","major_comments":[{"comment":"Construction (the blow-up step following Ciocan-Fontanine--Kim): the loci of linear dependence are taken as blow-up centers, yet no explicit computation or reference verifies that these centers have the expected codimension, that the resulting space remains irreducible, or that it satisfies the precise hypotheses (mild singularities, quasimap data form) of Farkas' purity theorem. This verification is load-bearing for the claim that the spaces are pure of expected dimension and that ordinary intersection theory yields enumerative counts.","section":"Construction section"}],"minor_comments":[{"comment":"The abstract states that the conjecture is proven in dimension 2 but does not name the Brill-Noether theorem as the main input; adding this would improve immediate clarity.","section":"Abstract"},{"comment":"Notation for the linear-dependence loci and the resulting exceptional divisors could be introduced with a displayed equation or short diagram to aid readability.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on the manuscript. We respond point by point to the major comment below.","responses":[{"response":"We agree that the manuscript does not currently contain an explicit verification of the codimension of the linear dependence loci or a direct confirmation that the blown-up space meets all hypotheses of Farkas' purity theorem. In the revision we will add a paragraph in the Construction section supplying these details. The codimension computation will follow from the expected rank of the evaluation map on the Ciocan-Fontanine--Kim space, using the same bundle data as in the projective-space case. Irreducibility of the resulting space will be argued by observing that the centers lie in the boundary divisor and the main component remains dominant after blow-up. For Farkas' theorem we will note that the blow-up centers are regularly embedded and that the quasimap data (including the sections and the stability condition) are preserved, yielding at worst mild singularities compatible with the theorem's hypotheses. This addition will make the purity claim fully grounded.","revision_made":"yes","referee_comment":"[Construction section] Construction (the blow-up step following Ciocan-Fontanine--Kim): the loci of linear dependence are taken as blow-up centers, yet no explicit computation or reference verifies that these centers have the expected codimension, that the resulting space remains irreducible, or that it satisfies the precise hypotheses (mild singularities, quasimap data form) of Farkas' purity theorem. This verification is load-bearing for the claim that the spaces are pure of expected dimension and that ordinary intersection theory yields enumerative counts."}],"tokens_in":1261,"tokens_out":357,"duration_ms":39671,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Hi, The key point with this paper is the construction of a moduli space of complete quasimaps to the blow-up of projective space, obtained by blowing up the standard quasimap space at linear dependence loci, together with a conjecture that its tautological intersections count curves of fixed complex structure on the blow-up. They do a good job extending the Ciocan-Fontanine-Kim framework in a natural way. The use of Farkas' purity result to control the dimension is a solid move, and the proof of the conjecture in dimension 2 via Brill-Noether theory on toric surfaces gives concrete evidence that the idea works at least in low dimensions. The abstract is straightforward about what is conjectural and what is proven. The potential issue is whether the blow-up step preserves the purity and the expected dimension without extra components. The stress-test note raises this, and it would be good to see an explicit check that the centers are of the right codimension and that the resulting space still meets the conditions for Farkas' theorem. Without that, the claim that the intersections are enumerative in the usual sense rests on an assumption that might need more justification in higher dimensions. The conjecture itself is not circular, as it introduces new objects. This work is for specialists in enumerative algebraic geometry who deal with moduli of curves and maps to projective varieties or their blow-ups. A reader looking for new compactifications that track complex structure more precisely would find value here. It has enough substance and novelty to merit a serious referee report rather than a desk rejection. I would recommend putting it through peer review.","headline":"This paper builds a new complete quasimap space to blow-ups of projective space by blowing up the standard quasimap space and conjectures that its intersections count fixed-complex-structure curves, with a proof only in dimension 2.","tokens_in":2336,"tokens_out":409,"would_cite":false,"duration_ms":31951,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Moduli blow-up construction at linear-dependence loci unrelated to RS cost or distinction forcing","alignment":"orthogonal","rationale":"The paper's core machinery is the iterated blow-up of augmented quasimap spaces Q⁺_β(C,X) along bi-rank strata (Definition 2.7.1), using Farkas purity (Prop. 2.3.3) and Brill-Noether transversality to obtain enumerative intersections on complete quasimaps to Bl_{P^s}(P^r). This is standard algebraic geometry of moduli spaces of maps; it invokes no recognition cost J(x), golden-ratio identities, φ-ladder spacings, 8-tick periodicity, or parameter-free derivations. No RS theorem (e.g., reality_from_one_distinction, J-uniqueness via Aczél, or Alexander-duality D=3 forcing) is paralleled or contradicted.","tokens_in":67751,"confidence":"high","tokens_out":199,"duration_ms":22173,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A moduli space of complete quasimaps to blown-up projective space is built by blowing up standard quasimaps at linear dependence loci, with the conjecture that its tautological intersections count curves of fixed complex structure under inc","keywords":["complete quasimaps","moduli spaces","blow-ups","enumerative geometry","tautological intersections","Brill-Noether theorem","toric surfaces"],"falsifier":"A concrete mismatch, in dimension three or higher, between the tautological intersection numbers on the complete quasimap space and independently known enumerative counts of fixed-complex-structure curves on the blown-up target.","tokens_in":2539,"feed_emoji":"","tokens_out":673,"duration_ms":30487,"temperature":0.7,"pith_summary":"The paper constructs moduli spaces of complete quasimaps to the blow-up Bl of P^r by P^s. The construction blows up the Ciocan-Fontanine-Kim quasimap space at the loci where sections of the tautological line bundles become linearly dependent. The central conjecture states that the tautological intersection numbers on these spaces enumerate curves of fixed complex structure on the target satisfying general incidence conditions. This differs from classical moduli spaces of maps, which permit the complex structure on the curve to vary. Purity of the expected dimension follows from a theorem of Farkas, and the conjecture is established in dimension two by invoking a Brill-Noether theorem for general curves on toric surfaces.","feed_headline":"Complete quasimaps count fixed-structure curves on blown-up spaces","feed_subtitle":"Tautological intersections on the new moduli spaces enumerate curves with prescribed complex structure under general incidence conditions.","key_machinery":"The moduli space of complete quasimaps, constructed by blowing up the Ciocan-Fontanine-Kim quasimap space at loci where sections of the relevant line bundles are linearly dependent.","core_discovery":"The authors define complete quasimaps to Bl_{P^s}(P^r) by successive blow-ups of the quasimap space at points of linear dependence among sections. They conjecture that the tautological intersections on the resulting moduli spaces compute enumerative counts of curves with fixed complex structure on the blown-up space, subject to general incidence conditions, and prove this conjecture when the target has dimension two.","pith_inferences":["The same blow-up procedure could be applied to quasimaps into other toric targets to obtain analogous enumerative theories.","Comparison with stable-map or stable-quasimap compactifications may reveal how fixing the complex structure alters the resulting curve counts.","Higher-dimensional cases would become accessible once Brill-Noether-type statements are established for curves on higher-dimensional toric varieties."],"forward_implications":["The spaces provide a compactification in which the complex structure on the domain curve stays fixed while still allowing intersection theory.","Tautological classes on these spaces directly yield enumerative invariants for curves on blown-up projective spaces.","In dimension two the conjecture supplies explicit counts via the Brill-Noether theorem on toric surfaces."],"fun_headline_variants":["Quasimap blow-ups count fixed complex structure curves","Tautological numbers count curves on complete quasimap spaces","Blown-up projective quasimaps enumerate fixed-structure curves","New blowup quasimaps give curve counts with fixed structure"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The blown-up moduli spaces remain pure of the expected dimension.","fun_headline_variants_meta":{"raw":{"variants":["Quasimap blow-ups count fixed complex structure curves","Tautological numbers count curves on complete quasimap spaces","Blown-up projective quasimaps enumerate fixed-structure curves","New blowup quasimaps give curve counts with fixed structure"]},"model":"grok-4.3","cost_usd":0.006312,"raw_usage":{"total_tokens":2926,"prompt_tokens":587,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":63124500,"prompt_tokens_details":{"text_tokens":587,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2274,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":587,"tokens_out":65,"duration_ms":21666,"temperature":1.0,"reasoning_tokens":2274,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-22T13:43:11.556210+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete mismatch, in dimension three or higher, between the tautological intersection numbers on the complete quasimap space and independently known enumerative counts of fixed-complex-structure curves on the blown-up target.","supporting_citations":[],"review_version":1}