{"id":"fee41219-6b67-4d5a-8821-841ea7363413","arxiv_id":"2606.30386","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A local-cohomology replacement for a depth-three step yields non-rational components of Hilb(A^n) for all n ≥ 10 in characteristic zero.","lead":"This paper gives a one-variable version of a prior construction showing that Hilbert schemes of points in affine space have non-rational irreducible components. The variant lowers the known threshold from dimension 12 to dimension 10 over fields of characteristic zero.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Sufficiency of low-degree HR-module vanishing to replace depth-three step in TNT frame","rationale":"The reader's weakest_assumption pinpoints exactly the unverified substitution step that carries the dimension reduction; no other part of the argument (the one-variable frame itself or the base curve) appears more fragile once that step is granted.","tokens_in":1645,"tokens_out":322,"duration_ms":28021,"concrete_test":"In the section presenting the local-cohomology replacement, recompute the relevant local-cohomology groups (or the spectral-sequence page that feeds the tangent-space formula) for the explicit g^3_9 curve; if any non-vanishing term remains that was previously killed by the depth-three hypothesis, the negativity claim for n=10 does not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The extension from n≥12 to n≥10 rests on replacing Jelisiejew's depth-three negativity computation with a local-cohomology argument that uses only the vanishing of H^1_* (I_C) and H^2_* (I_C) for the complete g^3_9 curve C. If this vanishing controls only the middle cohomology but leaves a non-vanishing contribution in the local-cohomology spectral sequence that affects the tangent space to the frame or the obstruction space, the required negative tangent direction may fail to appear at the lower ambient dimension. The abstract states the replacement occurs but supplies no explicit identification of which graded pieces of the local-cohomology module are killed and which survive.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to give a one-variable variant of the Farkas-Pandharipande-Sammartano construction of non-rational irreducible components of Hilb(A^n). By adjoining only one auxiliary variable and replacing the depth-three step of Jelisiejew's TNT frame with a local-cohomology argument that uses vanishing of the low-degree Hartshorne-Rao module of a complete g^3_9 curve, the authors extend the known range from n ≥ 12 to n ≥ 10 over a field of characteristic zero.","tokens_in":1774,"tokens_out":419,"duration_ms":34645,"significance":"If the local-cohomology replacement is valid, the result lowers the dimension threshold at which non-rational components of Hilb(A^n) are known to exist. The one-variable approach is more economical than the prior two-variable doubling construction and, if fully verified, would constitute a modest but concrete advance in the study of the geometry of Hilbert schemes of points.","major_comments":[{"comment":"The central technical claim—that vanishing of H^1_*(I_C) and H^2_*(I_C) for the complete g^3_9 curve C suffices to replace the depth-three negativity computation and still produce a negative tangent direction at ambient dimension n=10—requires an explicit identification of the surviving graded pieces of the local-cohomology module and a verification that they do not interfere with the tangent or obstruction spaces in the frame construction. The abstract states that the replacement occurs but supplies no such graded-piece computation or spectral-sequence analysis.","section":"local-cohomology replacement argument (Introduction and main construction)"}],"minor_comments":[{"comment":"The abstract refers to 'the vanishing of the low-degree Hartshorne--Rao module' without specifying the precise degrees or the exact curve; this should be stated explicitly in the introduction for clarity.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for acknowledging the potential significance of the one-variable construction. We address the single major comment below.","responses":[{"response":"We agree that the current exposition would benefit from greater explicitness on this point. The vanishing of the low-degree Hartshorne–Rao module is invoked to ensure that local-cohomology contributions do not produce positive-dimensional tangent directions at n=10, but the manuscript does not display the graded pieces or the relevant spectral-sequence pages. In the revised version we will add a short subsection (placed after the statement of the main construction) that (i) recalls the minimal free resolution of the ideal of the complete g^3_9 curve, (ii) computes the dimensions of the surviving graded summands of H^3_*(I_C) in the degrees that interact with the frame, and (iii) shows via the local-to-global Ext spectral sequence that these summands map to zero in the tangent space of the Hilbert scheme at the chosen point when the ambient dimension is 10. This computation uses only the known vanishing and the Castelnuovo–Mumford regularity of the curve; no new geometric input is required.","revision_made":"yes","referee_comment":"The central technical claim—that vanishing of H^1_*(I_C) and H^2_*(I_C) for the complete g^3_9 curve C suffices to replace the depth-three negativity computation and still produce a negative tangent direction at ambient dimension n=10—requires an explicit identification of the surviving graded pieces of the local-cohomology module and a verification that they do not interfere with the tangent or obstruction spaces in the frame construction. The abstract states that the replacement occurs but supplies no such graded-piece computation or spectral-sequence analysis."}],"tokens_in":1303,"tokens_out":394,"duration_ms":40362,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's main advance is a one-variable frame that reaches non-rational components in Hilb(A^n) for n=10 and 11, where the earlier Farkas-Pandharipande-Sammartano work stopped at 12. It keeps the same curve source but replaces the depth-three negativity step in Jelisiejew's TNT construction with a local-cohomology argument that uses only the vanishing of the low-degree Hartshorne-Rao module.\n\nThat replacement is the concrete new piece. The rest follows the two-variable template closely: start from the curve in P^3, adjoin one extra variable instead of two, apply the frame, and avoid the doubling. The vanishing input is standard for that particular complete linear series, so the setup is plausible on its face.\n\nThe soft spot is whether the H^1 and H^2 vanishing really supplies the full negative tangent direction once the ambient dimension drops. The local-cohomology spectral sequence could leave extra terms in the obstruction space or the tangent space to the frame, and the abstract does not identify the surviving graded pieces. If the full derivation checks that those terms do not interfere, the extension holds; otherwise the n=10 claim rests on an incomplete substitution.\n\nThis is a short technical note aimed at people already working on rationality questions for Hilbert schemes of points. A reader who knows the cited papers will see the incremental change immediately. It is worth sending to referees because the claim is precise, the method is a direct variant, and the potential gap is narrow enough that a careful check can settle it.","headline":"Wu lowers the known threshold for non-rational components of Hilb(A^n) from 12 to 10 by adapting the prior construction to one variable via local cohomology on the g^3_9 curve.","tokens_in":2207,"tokens_out":410,"would_cite":false,"duration_ms":43728,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Over fields of characteristic zero, Hilb(A^n) has non-rational irreducible components for all n at least 10.","keywords":["Hilbert schemes of points","non-rational components","local cohomology","Hartshorne-Rao module","g^3_9 curve","frame construction","affine space"],"falsifier":"An explicit computation showing that the low-degree Hartshorne-Rao module of the complete g^3_9 curve does not vanish would prevent the local-cohomology replacement from working and thereby block the construction for n=10.","tokens_in":2544,"feed_emoji":"","tokens_out":737,"duration_ms":55719,"temperature":0.7,"pith_summary":"The paper lowers the dimension threshold at which the Hilbert scheme of points in affine space is known to have non-rational components. Earlier constructions reached only n at least 12 by adjoining two auxiliary variables to a curve Hilbert scheme and applying a frame construction twice. This version adjoins only one auxiliary variable and replaces the depth-three step in the tangent-space computation with a local-cohomology argument that uses the vanishing of the low-degree Hartshorne-Rao module on a complete g^3_9 curve. A reader would care because the result shows that irrationality appears in point configurations already in ten-dimensional space rather than twelve.","feed_headline":"Non-rational components of point Hilbert schemes reach n=10","feed_subtitle":"One-variable construction with local cohomology lowers the threshold from 12 over characteristic zero.","key_machinery":"one-variable frame construction that replaces the depth-three step in Jelisiejew's negative tangent computation with a local-cohomology argument based on the vanishing of the low-degree Hartshorne-Rao module for the complete g^3_9 curve source","core_discovery":"Over a field of characteristic zero, Hilb(A^n) has non-rational irreducible components for all n greater than or equal to 10. The proof proceeds by a one-variable frame construction: start from the Hilbert scheme of a complete g^3_9 curve in P^3, adjoin one auxiliary variable, apply a local-cohomology replacement for the depth-three step in Jelisiejew's negative tangent computation that relies on the vanishing of the low-degree Hartshorne-Rao module, and then double the number of variables.","pith_inferences":["The same local-cohomology replacement could be tested on other curve sources whose Hartshorne-Rao modules vanish in low degrees.","The threshold for known non-rationality is now at most 10 rather than 12, narrowing the range where rationality remains possible."],"forward_implications":["Non-rational irreducible components exist in Hilb(A^n) for n=10 and n=11 as well as for all larger n.","The frame construction succeeds when only one auxiliary variable is adjoined rather than two.","The local-cohomology argument produces the required negative tangent space once the Hartshorne-Rao vanishing holds.","Doubling the number of variables after the one-variable step yields the result for every n at least 10."],"fun_headline_variants":["One-variable frame lowers non-rational Hilb threshold to n=10","One aux var reaches non-rational point Hilb components at n=10","Local coh variant drops Hilb non-rational threshold to n=10","One-var construction hits non-rational Hilb components at n=10"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The vanishing of the low-degree Hartshorne-Rao module for the complete g^3_9 curve source is sufficient to replace the depth-three step in Jelisiejew's negative tangent computation with a local-cohomology argument.","fun_headline_variants_meta":{"raw":{"variants":["One-variable frame lowers non-rational Hilb threshold to n=10","One aux var reaches non-rational point Hilb components at n=10","Local coh variant drops Hilb non-rational threshold to n=10","One-var construction hits non-rational Hilb components at n=10"]},"model":"grok-4.3","cost_usd":0.007392,"raw_usage":{"total_tokens":3383,"prompt_tokens":637,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":73924500,"prompt_tokens_details":{"text_tokens":637,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2668,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":637,"tokens_out":78,"duration_ms":46847,"temperature":1.0,"reasoning_tokens":2668,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T03:30:20.438620+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation showing that the low-degree Hartshorne-Rao module of the complete g^3_9 curve does not vanish would prevent the local-cohomology replacement from working and thereby block the construction for n=10.","supporting_citations":[],"review_version":1}