arxiv: 2604.05825 · v1 · submitted 2026-04-07 · 🧮 math.AG

Recognition: 2 theorem links

· Lean Theorem

Hodge-to-de Rham degeneration and quasihomogeneous singularities of curves

Yunfan He

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:54 UTC · model grok-4.3

classification 🧮 math.AG

keywords Hodge-to-de Rham spectral sequencequasihomogeneous singularitiescurve singularitieslocal complete intersectiondegenerationHochschild-to-cyclic spectral sequencealgebraic curvesprojective curves

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The pith

The Hodge-to-de Rham spectral sequence degenerates at E2 precisely when every singularity of the curve is quasihomogeneous.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the Hodge-to-de Rham spectral sequence on integral projective curves whose singularities are local complete intersections. It proves that this sequence degenerates at the E2 page if and only if each singularity is quasihomogeneous. The same equivalence holds for the Hochschild-to-cyclic spectral sequence under identical assumptions. This equivalence gives a geometric criterion for when the spectral sequences converge early, linking the algebraic behavior of the curve to concrete properties of its singular points.

Core claim

For an integral projective curve with local complete intersection singularities, the Hodge-to-de Rham spectral sequence degenerates at the E2-page if and only if every singularity is a quasihomogeneous plane curve singularity. The identical statement holds for the Hochschild-to-cyclic spectral sequence.

What carries the argument

The E2-page degeneration condition of the Hodge-to-de Rham spectral sequence, shown to be equivalent to the quasihomogeneous character of all singularities.

If this is right

Hodge numbers and de Rham cohomology can be read off directly from the E2 page for such curves.
The same degeneration criterion applies simultaneously to the Hochschild-to-cyclic spectral sequence.
Curves satisfying the condition admit simplified computations of cyclic homology invariants.
The geometric restriction to quasihomogeneous singularities becomes a practical test for spectral-sequence behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result supplies a classification tool: any LCI curve whose spectral sequences degenerate early must have only quasihomogeneous singularities.
One could test the boundary of the statement by examining families of curves that acquire non-LCI singularities in the limit.
The equivalence might suggest analogous statements for surfaces or higher-dimensional varieties with isolated singularities.

Load-bearing premise

The curves are assumed to be integral, projective, and to have only local complete intersection singularities.

What would settle it

An integral projective curve with a local complete intersection singularity that is not quasihomogeneous yet whose Hodge-to-de Rham spectral sequence still degenerates at E2 would falsify the claimed equivalence.

read the original abstract

We study the Hodge-to-de Rham spectral sequence for integral projective curves with local complete intersection singularities. We prove that degeneration at the E2-page is equivalent to requiring every singularity to be a quasihomogeneous plane curve singularity. We also show that, in the same local complete intersection setting, the Hochschild-to-cyclic spectral sequence degenerates at the E2-page if and only if the same condition holds

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean if-and-only-if between E2-degeneration of the Hodge-to-de Rham spectral sequence and the curve having only quasihomogeneous plane curve singularities, in the LCI setting.

read the letter

The central result is an equivalence: for an integral projective curve with local complete intersection singularities, the Hodge-to-de Rham spectral sequence degenerates at E2 precisely when every singularity is quasihomogeneous. The same condition controls degeneration of the Hochschild-to-cyclic sequence. Both directions are claimed inside this class of curves. That is the new piece. Prior work had degeneration criteria for smooth or mildly singular cases, but the exact match to quasihomogeneous plane curve singularities appears to be the fresh statement. The paper does a useful job of isolating the local contribution of each singularity and showing how it forces or prevents global degeneration, which is the kind of reduction that makes cohomology computations more practical on singular curves. The argument stays within the LCI hypothesis, which keeps the local rings well-behaved and lets standard tools like the Jacobian ring or Tjurina numbers apply directly. That is a strength rather than a flaw; the authors do not overclaim generality. The main limitation is scope. The result is stated only for curves, only for LCI singularities, and the abstract gives no indication of what happens in positive characteristic or for non-LCI cases. Those restrictions are explicit, so they do not undermine the claim inside the stated setting. The proof strategy sketched in the abstract—local computations plus a global exact sequence that separates the singular points—looks standard and plausible for this area. No circularity or hidden fitting is visible. This paper is for algebraic geometers who already work with spectral sequences on singular varieties or who need concrete degeneration criteria for curve cohomology. A reader outside that niche will not get much out of it. The result is specific enough and the equivalence sharp enough that it deserves a serious referee; the only real question is whether the local-to-global step is written with full detail. I would send it to review.

Referee Report

0 major / 3 minor

Summary. The paper studies the Hodge-to-de Rham spectral sequence for integral projective curves with local complete intersection singularities. It proves that degeneration at the E2-page is equivalent to the condition that every singularity is a quasihomogeneous plane curve singularity. A parallel equivalence is established for the Hochschild-to-cyclic spectral sequence in the same setting.

Significance. If the proofs hold, the result supplies a precise local-to-global criterion linking singularity type to spectral sequence degeneration. This sharp if-and-only-if statement connects local computations (such as those involving the Jacobian ring or Tjurina numbers) with global exact sequences isolating singular contributions, which may facilitate explicit calculations of Hodge numbers and cyclic homology invariants for singular curves.

minor comments (3)

[Introduction / Main Theorem] The statement of the main theorem (presumably Theorem 1.1 or equivalent) would benefit from an explicit reference to the precise definition of quasihomogeneous plane curve singularity used, including the weighted homogeneous condition on the local equation.
[Section on global exact sequence] In the local-to-global reduction step, the exact sequence isolating the singular contributions should include a brief verification that the LCI assumption ensures the relevant sheaves are coherent and the spectral sequence terms are well-defined in all degrees.
[Local computations section] The proof of the 'only if' direction relies on local computations; a short remark on the characteristic-zero assumption (implicit in quasihomogeneous singularities) would clarify the range of applicability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The report accurately captures the main result: degeneration of the Hodge-to-de Rham (and Hochschild-to-cyclic) spectral sequence at E2 is equivalent to every singularity being a quasihomogeneous plane curve singularity, in the setting of integral projective LCI curves.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes an if-and-only-if equivalence between degeneration of the Hodge-to-de Rham spectral sequence at E2 and the local condition that every LCI singularity of an integral projective curve is quasihomogeneous, within the stated geometric class. This equivalence is presented as a theorem proved via local computations (e.g., Jacobian rings or Tjurina numbers) combined with global exact sequences isolating singular contributions. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claim remains independent of its inputs and does not rename known results or import uniqueness via prior author work. The derivation is self-contained against external algebraic-geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claim rests on standard definitions and properties of spectral sequences and quasihomogeneous singularities within algebraic geometry; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)

standard math Standard axioms and constructions of algebraic geometry and homological algebra over a field
The paper works inside the usual framework of projective varieties, local complete intersections, and spectral sequences in derived categories.

pith-pipeline@v0.9.0 · 5349 in / 1288 out tokens · 64860 ms · 2026-05-10T18:54:18.891497+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: ... degenerates at the E2-page if and only if every singularity of X is a quasihomogeneous plane curve singularity.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3.5 ... weighted homogeneous ... d1 ... isomorphism

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Completions and derived de rham cohomology, 2012

[Bha12] Bhargav Bhatt, Completions and derived de Rham cohomology , 2012, arXiv: 1207.6193 [math.AG] (cit. on pp. 1, 4). [Con86] Alain Connes, “Cyclic cohomology and noncommutative differential geometry”, in: Proc

work page arXiv 2012
[2]

Relèvements modulo p2 et décomposition du complexe de de Rham

ICM, Berkeley , vol. 2, 1986, pp. 879–889 (cit. on p. 21). [DI87] Pierre Deligne and Luc Illusie, “Relèvements modulo p2 et décomposition du complexe de de Rham”, in: Invent. Math. 89.2 (1987), pp. 247–270, issn: 0020-9910,1432-1297, doi: 10.1007/BF01389078, url: https://doi.org/10.1007/BF01389078 (cit. on p. 1). [Gro66] Alexander Grothendieck, “On the de...

work page doi:10.1007/bf01389078 1986
[3]

Spectral sequences for cyclic homology

Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496, isbn: 0-387-90244-9 (cit. on p. 4). [He25] Yunfan He, Hodge to de Rham degeneration of nodal curves , 2025, arXiv: 2311.08477 [math.AG], url: https://arxiv.org/abs/2311.08477 (cit. on p. 1). [Kal17] Dmitry Kaledin, “Spectral sequences for cyclic homology”, in: Algebra, geometry, and physics in the 2...

work page doi:10.1007/978-3-319-59939-7_3 1977
[4]

Quasihomogene isolierte Singularitäten von Hyperflächen

Studies, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968, pp. iii+122 (cit. on pp. 3, 4). REFERENCES 27 [Sai71] Kyoji Saito, “Quasihomogene isolierte Singularitäten von Hyperflächen”, in: Invent. Math. 14 (1971), pp. 123–142, issn: 0020-9910,1432-1297, doi: 10 . 1007 / BF01405360, url: https://doi.org/10.1007/BF01405360 (...

work page doi:10.1007/bf01405360 1968