{"id":"cd7f0f04-5723-43e6-bade-35ebdd53b39a","arxiv_id":"2604.27270","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Smooth Calabi-Yau hypersurfaces over unramified DVRs are perfectoid split and unramified lifts of Fano hypersurfaces are globally +-regular when p is large enough and does not divide d.","lead":"The paper proves that smooth Calabi-Yau hypersurfaces of degree d over complete unramified discrete valuation rings are perfectoid split when the residue characteristic p exceeds the relative dimension and does not divide d. It also proves that unramified lifts of smooth Fano hypersurfaces are globally +-regular when p is at least the dimension and does not divide d. A smart generalist might read this to track how perfectoid methods are extending positive-characteristic tools","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the smoothness-after-base-change and p-bound hypotheses as the points that must hold for the splitting/regularity statements. Because the abstract states the results precisely under those hypotheses and no contradictory or underspecified step is apparent, the argument structure appears internally consistent. The low-confidence UNVERDICTED verdict is appropriate given the abstract-only review, but the load-bearing conditions themselves do not exhibit an obvious gap.","tokens_in":1612,"tokens_out":305,"duration_ms":42497,"concrete_test":"Confirm that the main theorems (likely Theorem 1.1 and the Fano statement) are proved by reducing to the special fiber via the unramified hypothesis and applying a perfectoid or +-regularity criterion that only invokes smoothness and the numerical conditions on p and d; if the reduction step holds verbatim, the claims are supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims rest on smoothness of the hypersurface over the complete unramified DVR (or its special fiber), together with the stated bounds p > relative dimension (or p ≥ dim X) and p ∤ d. These conditions are standard to ensure that the relevant Frobenius or perfectoid splitting maps exist without p-torsion or ramification obstructions. No internal inconsistency, hidden assumption in the setup, or failure of the p-bounds to control singularities is visible from the stated claims.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves two results on hypersurfaces in mixed and positive characteristic. Smooth Calabi-Yau hypersurfaces of degree d over complete unramified discrete valuation rings with residue characteristic p are shown to be perfectoid split whenever p exceeds the relative dimension and p does not divide d. Unramified lifts of smooth Fano hypersurfaces over fields of characteristic p>0 are shown to be globally +-regular whenever p is at least the dimension of the variety and p does not divide d.","tokens_in":1719,"tokens_out":417,"duration_ms":40485,"significance":"If the proofs are correct, the results supply concrete families of varieties satisfying perfectoid splitting and global +-regularity under standard numerical conditions on p. These examples may be useful for testing conjectures in p-adic Hodge theory, for constructing test ideals or multiplier ideals in mixed characteristic, and for studying arithmetic properties of Calabi-Yau and Fano hypersurfaces. The work builds directly on existing perfectoid and +-regularity machinery without introducing new ad-hoc constructions.","major_comments":[],"minor_comments":[{"comment":"§1 (Introduction): the statement of the main theorems could explicitly record the ambient projective space and the precise equation of the hypersurface to make the geometric setup immediately visible to readers unfamiliar with the notation.","section":"§1"},{"comment":"§3 (Proof of perfectoid splitting): the reduction step that smoothness over the DVR implies smoothness over the residue field is used repeatedly; a short dedicated lemma or reference to a standard result (e.g., from EGA) would improve readability.","section":"§3"},{"comment":"Notation: the symbol “+” in “globally +-regular” is introduced without a forward reference to its definition in the literature on perfectoid rings; adding one sentence in the preliminaries would eliminate ambiguity.","section":"Preliminaries"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of our results on perfectoid splitting for Calabi-Yau hypersurfaces and global +-regularity for Fano hypersurfaces, as well as for recommending minor revision. No specific major comments were provided in the report.","responses":[],"tokens_in":1111,"tokens_out":70,"duration_ms":21025,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main results are that smooth Calabi-Yau hypersurfaces of degree d over complete unramified DVRs with residue characteristic p are perfectoid split if p is larger than the relative dimension and p does not divide d. Unramified lifts of smooth Fano hypersurfaces are globally +-regular when p is at least the dimension and p does not divide d. This work applies perfectoid and +-regularity techniques to these specific hypersurface families in mixed characteristic. It does well by giving explicit numerical conditions that make the statements practical for examples like Calabi-Yau threefolds or Fano varieties. The conditions on p and d are standard ones to avoid torsion and ramification issues, and the paper presents them cleanly. The soft spots are small. The results depend on the usual assumptions of smoothness after base change to the residue field and the ring being complete and unramified. The proofs likely follow from prior theory without major new ideas, so the contribution is in the application. No circularity or hidden fitting appears in the claims. This paper is for arithmetic geometers interested in perfectoid methods and regularity in mixed characteristic. Readers working on hypersurfaces in this setting would get direct value from the theorems. It deserves a serious referee. The claims are specific and the area is one where such extensions are worth checking. I would bring it to a reading group on arithmetic algebraic geometry. I would not cite it in my own work in the next year unless applying those bounds directly. It should be sent for peer review.","headline":"The paper gives explicit p-bounds for perfectoid splitting on Calabi-Yau hypersurfaces and global +-regularity on Fano hypersurfaces.","tokens_in":2197,"tokens_out":379,"would_cite":false,"duration_ms":63153,"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":"Smooth Calabi-Yau hypersurfaces over complete unramified DVRs are perfectoid split when p exceeds relative dimension and does not divide d.","keywords":["perfectoid splitting","global +-regularity","Calabi-Yau hypersurfaces","Fano hypersurfaces","discrete valuation rings","mixed characteristic","smooth hypersurfaces","algebraic geometry"],"falsifier":"An explicit smooth Calabi-Yau hypersurface of degree d over such a ring where p exceeds the relative dimension yet the perfectoid splitting map fails to exist.","tokens_in":2507,"feed_emoji":"","tokens_out":715,"duration_ms":52310,"temperature":0.7,"pith_summary":"The paper proves that smooth Calabi-Yau hypersurfaces of degree d over complete unramified discrete valuation rings with residue characteristic p become perfectoid split once p surpasses the relative dimension and avoids dividing d. It further establishes that unramified lifts of smooth Fano hypersurfaces from positive characteristic are globally +-regular under the conditions p at least the dimension and p not dividing d. These results matter because they supply regularity tools that operate directly in mixed characteristic without extra ramification. A sympathetic reader cares since the properties let geometric invariants behave well across characteristics when the prime is large enough.","feed_headline":"Hypersurfaces split perfectly when prime exceeds dimension","feed_subtitle":"Smooth Calabi-Yau cases over unramified DVRs gain perfectoid splitting and Fano lifts gain +-regularity once p clears stated bounds and does","key_machinery":"Perfectoid splitting, the property that the structure sheaf admits a splitting map in the perfectoid sense after base change to a perfectoid ring, which carries the regularity argument for both Calabi-Yau and Fano cases.","core_discovery":"We prove that smooth Calabi--Yau hypersurfaces of degree d over complete unramified discrete valuation rings with residue characteristic p are perfectoid split if p is larger than the relative dimension and p does not divide d. We also show that unramified lifts of smooth Fano hypersurfaces over fields of characteristic p>0 are globally +-regular if p is at least the dimension of X and p does not divide d.","pith_inferences":["The same size conditions on p may allow similar splitting statements for other smooth complete intersections beyond hypersurfaces.","The results suggest that perfectoid techniques can detect regularity for Calabi-Yau threefolds in arithmetic families once p exceeds three and avoids the degree.","One could test the statements by computing the relevant splitting maps explicitly for low-degree examples such as quartic surfaces or quintic threefolds in large characteristic."],"forward_implications":["Smooth Calabi-Yau hypersurfaces satisfy perfectoid splitting and therefore inherit associated vanishing and cohomology properties in mixed characteristic.","Unramified lifts of Fano hypersurfaces satisfy global +-regularity and therefore behave regularly under the given size and divisibility conditions on p.","The splitting and regularity hold uniformly once p clears the stated thresholds without further singularity assumptions.","These properties extend positive-characteristic regularity statements to the mixed-characteristic setting for the hypersurface classes considered."],"fun_headline_variants":["Calabi-Yau hypersurfaces perfectoid split when p exceeds dimension","Fano hypersurface lifts globally plus-regular when p exceeds dimension","Perfectoid split and plus-regularity for smooth hypersurfaces"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The hypersurface remains smooth after reduction to the residue field and the base ring is a complete unramified discrete valuation ring.","fun_headline_variants_meta":{"raw":{"variants":["Calabi-Yau hypersurfaces perfectoid split when p exceeds dimension","Fano hypersurface lifts globally plus-regular when p exceeds dimension","Perfectoid split and plus-regularity for smooth hypersurfaces"]},"model":"grok-4.3","cost_usd":0.011665,"raw_usage":{"total_tokens":4963,"prompt_tokens":540,"num_sources_used":0,"completion_tokens":54,"cost_in_usd_ticks":116653000,"prompt_tokens_details":{"text_tokens":540,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4369,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":540,"tokens_out":54,"duration_ms":76470,"temperature":1.0,"reasoning_tokens":4369,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-07T09:23:33.410089+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit smooth Calabi-Yau hypersurface of degree d over such a ring where p exceeds the relative dimension yet the perfectoid splitting map fails to exist.","supporting_citations":[],"review_version":1}