{"id":"8d47df51-1608-4b3a-9d94-16867c55b944","arxiv_id":"2504.12402","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Higher Du Bois singularities are shown to be equivalent, in many characteristic-zero settings, to K-regularity, yielding a strengthened Vorst conjecture for local complete intersections and a projective characterization of K-regularity.","lead":"This paper proves that a Hodge-theoretic measure of singularities, higher Du Bois conditions, controls when the algebraic K-groups of a variety stop changing after multiplying by a line. It uses this to show that affine local complete intersections which are K_2-regular are actually smooth, and it gives numerical criteria in terms of the minimal exponent.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.1's base-change step is not established; Proposition 7.3's diagram omits the surjectivity that would make it work.","rationale":"The reader's weakest assumption correctly identifies the descent/base-change step in Theorem 7.1 as fragile, and the manuscript itself admits the direct Q-level implication is unclear. My stress-test sharpens this: the roundabout argument relies on Proposition 7.3, and the proof of Proposition 7.3 as printed does not license the conclusion without an additional unstated surjectivity. The cuspidal-cubic check shows the most naive way to fill the gap is false, so the issue is not merely expository. I do not move the verdict because the reader already made it CONDITIONAL, and because the paper's headline affine Vorst-strengthening is proved independently of Theorem 7.1. The concern is therefore real but localized: it affects Theorem I and the projective examples, not the main regularity criteria for affine local complete intersections. No evidence of circularity or dishonesty was found; the gap is a missing justification in a technically dense spectral-sequence argument.","tokens_in":28862,"tokens_out":23984,"duration_ms":248045,"concrete_test":"Re-derive Proposition 7.3 with all arrows in the commutative square made explicit, and verify whether the claimed surjectivity of H^i(X,LΩ≤p) -> H^i(X,Ω≤p) follows from the displayed Hodge-to-de Rham degeneration alone. In particular, test the p=0 case for a cuspidal projective cubic: compute H^1(O_X) and H^1(Ω^0_X) explicitly; if the map between them is not surjective, Proposition 7.3 is false for p=0 and Theorem 7.1 collapses. If it is surjective, identify the precise theorem in [Bh] or Du Bois that supplies the missing surjectivity, then run the same check for the component of Example 4.3 that uses Theorem 7.1 with m=-1 to confirm the asserted isomorphisms hold numerically.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The projective characterization (Theorem 7.1) is the least secure part of the paper. The proof first obtains the desired isomorphisms over Q and then, in the footnote, explicitly says it is unclear that these imply the maps over the original field F; the 'roundabout' spectral-sequence route that follows is used to bridge Q to F. The bridge depends on Proposition 7.3. As written, Proposition 7.3's proof is incomplete: after reducing to F=C, it displays a square with top row H^i(X,C) -> H^i(X,Ω≤p) (surjective by Hodge degeneration) and concludes that the map H^i(X,LΩ≤p) -> H^i(X,Ω≤p) is surjective. That inference requires an additional arrow in the square to be a surjection or isomorphism; the proof does not state which. The natural candidate, surjectivity of H^i(X,dR) -> H^i(X,LΩ≤p), fails already for p=0 for a cuspidal projective cubic: H^1(dR)=0 while H^1(O_X)=H^1(LΩ≤0)=1. Thus a missing argument is essential, not cosmetic. Since Examples 4.3 and 4.5 invoke Theorem 7.1, they inherit this gap. The affine Vorst-strengthening (Theorems B and F, Corollaries E and 4.15) is proved via Proposition 4.7 and Corollary 4.14 and does not appear to rely on Theorem 7.1, so the gap is localized to the projective side.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a dictionary between higher Du Bois singularities and K_m-regularity for varieties over fields of characteristic zero. The main affine results are: Proposition C and Theorem D, which connect m-Du Bois/pre-m-Du Bois conditions to K-regularity in both directions; Corollary E, an equivalence for local complete intersections with isolated singularities; Theorem F, which upgrades Vorst's conjecture for affine local complete intersections by showing that K_2-regularity (or K_1-regularity, depending on the parity of the codimension of the singular locus) forces regularity; and Theorem A, a numerical criterion for hypersurfaces via the minimal exponent. The paper also contains a study of Bass's question using Du Bois invariants of surfaces, including a claim that all pairs (b_{0,1}, b_{1,1}) with b_{0,1} >= b_{1,1} occur, and a projective characterization of K_m-regularity (Theorem 7.1 = Theorem I) that is used in two examples.","tokens_in":29110,"tokens_out":16580,"duration_ms":149224,"significance":"If the results are correct, the paper makes a substantial contribution: it gives the first systematic bridge between higher Du Bois singularities and algebraic K-theory, proves a strong form of Vorst's conjecture for local complete intersections in characteristic zero, and produces new tools for constructing and detecting K_m-regular varieties. The affine Vorst strengthening is particularly valuable because it is proved using the established CHWW/CHSW machine plus vanishing theorems for Du Bois complexes, and it does not appear to depend on the projective Theorem 7.1. The paper also gives useful examples and a surface-level classification of Du Bois invariants. However, the projective characterization Theorem 7.1 is not established as written: its proof contains a missing base-change justification and an incomplete diagram argument, and the text itself flags the unclear step. Since Theorem 7.1 is stated as a main theorem and is used in Examples 4.3 and 4.5, the paper needs substantive repair before the projective statements can be accepted.","major_comments":[{"comment":"The proof of Theorem 7.1 is incomplete as written. The reduction to F=C in Proposition 7.3 is justified only by 'standard base change arguments' referring to Lemma 2.10; Lemma 2.10 provides spectral sequences relating the L^p and Ω^p objects over different fields, but no statement in the paper establishes that the surjectivity or isomorphy of the displayed hypercohomology maps over C descends to the original field F. Second, the commutative diagram in the proof of Proposition 7.3 omits the vertical maps: from the surjectivity of H^i(X,C) -> H^i(X,Ω^{≤p}_{X/C}) one cannot conclude that H^i(X,LΩ^{≤p}_{X/C}) -> H^i(X,Ω^{≤p}_{X/C}) is surjective. The natural candidate for the missing arrow, H^i(X,C) -> H^i(X,LΩ^{≤p}_{X/C}), fails in general: for a cuspidal projective cubic and p=0, H^1(X,C)=0 while H^1(X,O_X)=H^1(X,LΩ^{≤0}_{X/C}) is one-dimensional. The text itself notes in footnote 4 that the direct implication is unclear and takes a 'roundabout' spectral-sequence route, but that route rests on Proposition 7.3, so the missing justification is load-bearing. Consequently Theorem I, Remark 7.2, and the projective examples (Examples 4.3 and 4.5) are not established as written.","section":"§7 (Proposition 7.3 and Theorem 7.1)"},{"comment":"The proof of part (2) of Theorem F begins 'Suppose codim X_sing = 2m+1 is odd', even though the statement of part (2) assumes that codim X_sing is even. The displayed inequalities that follow (for example m ≤ codim X_sing - 2 = 2m - 2) are the even-case computation. As printed, the proof does not cover the stated even-codimension case; it should begin with codim X_sing = 2m and then recheck the inequalities, especially the boundary case m=1.","section":"§5 (proof of Theorem F, part (2))"},{"comment":"Proposition 6.3(2) is under-specified. The hypersurface is written as x^3+y^3+z^{3m}+xyz^{m+a} (or possibly with a separate additive constant a), but the parameter a is not defined, and the statement claims b_{1,1}=n while the proof concludes b_{1,1}=a. The proof also treats a=0 and a≥1 separately and invokes a deformation argument for the independence of b_{0,1} from a. Please define the family f_a precisely, state the relation between a and n (presumably a=n), and spell out the deformation argument before claiming the classification for all pairs m≥n≥0.","section":"§6 (Proposition 6.3(2))"}],"minor_comments":[{"comment":"The hypothesis -d+2m+2<0 appears only inside the proof of Corollary 4.8; it should be stated in the corollary, since the claimed equivalence uses Proposition 4.4(1), which requires m ≤ (d-3)/2.","section":"§4 (Corollary 4.8)"},{"comment":"In the isolated-singularities case, the proof says the local-to-global spectral sequence degenerates but does not justify why H^p(X,NK_q(X))=0 for p>0; this follows because NK_q is supported on the singular locus, and that justification should be given explicitly.","section":"§4 (proof of Proposition 4.1(2))"},{"comment":"The implication K_{-d+2m+s}-regular ⇒ m-Du Bois is stated in Remark 4.16 without repeating the affine local-complete-intersection hypotheses from Proposition 4.7; the hypotheses should be included to avoid overstatement.","section":"§4 (Remark 4.16)"},{"comment":"There are numerous rendering artifacts in the manuscript, including the string '/leftr⫯g⊸tl⫯ne →' in place of arrows in exact triangles and displayed equivalences, and several missing superscript braces in formulas such as z^{3m} and xyz^{m+a} in §6; these should be corrected in the final version.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The affine Vorst part of the paper appears sound and is independent of Theorem 7.1, so the core contribution may survive even if the projective characterization is weakened or postponed. The main risk is overclaiming Theorem I: the missing diagram argument in Proposition 7.3 and the unclear base-change step are not cosmetic. I did not find circularity: the K-theory side is imported from CHWW and the Du Bois side from the cited literature. A revised version that either proves Theorem 7.1 with a complete argument or explicitly marks it as conditional would be appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does something genuinely useful: it gives a systematic bridge between higher Du Bois singularities and K-regularity, and the affine part appears well-founded. The minimal exponent criterion (Theorem A), the Vorst strengthening for local complete intersections (Theorem F), and the surface Du Bois table results (Proposition G) are new and interesting. The paper is honest about its own soft points, cites the heavy machinery properly, and I found no circularity.\n\nThe weakness is concentrated in the projective characterization, Theorem 7.1. The proof leans on Proposition 7.3, whose argument as written does not close. After reducing to C, Hodge degeneration gives a surjection H^i(X,C) -> H^i(X, Ω≤p), but the conclusion that H^i(X,LΩ≤p) -> H^i(X,Ω≤p) is surjective needs an additional arrow that is neither stated nor shown. The obvious candidate, surjectivity of H^i(X,dR) -> H^i(X,LΩ≤p), is false for p=0 on a cuspidal projective cubic. The paper's own footnote says the direct implication is unclear and then takes a spectral-sequence detour, but that detour as written does not fill the gap; it just moves the missing surjectivity to a different input. This means Examples 4.3 and 4.5, which use Theorem 7.1, inherit the uncertainty.\n\nThe affine results do not appear to depend on Theorem 7.1. Theorems B, D, F, and Corollary E are proved via Proposition 4.7 and the external vanishing theorems, so the affine core is likely solid. There are also two smaller issues worth flagging: a parity typo in the proof of Theorem F(2) (it repeats 'odd' instead of 'even'), and Proposition 6.3(2) leaves the parameter a undefined in the statement—the proof later uses it but the reader has to guess.\n\nIn short: this deserves a serious referee. The affine part is a real advance and the projective gap is localized; it might be fixable with a more careful argument or a restricted statement. I would not accept it as-is, but I would send it out rather than desk-reject. A thoughtful referee could clear up the projective question and the minor issues, leaving a worthwhile paper.","headline":"A serious paper with a strong affine core and a plausible but under-supported projective characterization; the gap in Theorem 7.1's proof should keep it from being approved as-is.","tokens_in":29670,"tokens_out":2576,"would_cite":true,"duration_ms":26275,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-16T12:32:31.593064+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}