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arxiv: 2604.03111 · v2 · submitted 2026-04-03 · 🧮 math.AG

Hilbert scheme of points on non-reduced nodal curves

Yuze Luan This is my paper

Pith reviewed 2026-05-13 18:59 UTC · model grok-4.3

classification 🧮 math.AG
keywords Hilbert scheme of pointsnon-reduced curvesweak diagonal partitionPoincaré polynomialOblomkov-Rasmussen-Shende conjectureHopf linkstratificationplane curve singularity
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The pith

Filtrations and an adapted valuation stratify the punctual Hilbert scheme on x^u y^v=0 by weak diagonal partitions, yielding explicit Poincaré polynomials for several u and v.

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

The paper develops a combinatorial stratification for the space of ideals with fixed support on the non-reduced nodal curve x to the power u times y to the power v equals zero. Strata are indexed by weak diagonal partitions that arise from filtrations on ideals together with a valuation adapted to the non-reduced structure. For u equal to 1 or 2 the strata are affine spaces, and for u equal to v, v-1 or v-2 they are products of algebraic tori with affine spaces. This geometric control produces explicit formulas for the Poincaré polynomials of the Hilbert scheme in those ranges. The formulas are then used to verify the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link when u is 1 and v is arbitrary.

Core claim

We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, x^u y^v=0. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on introducing filtrations on ideals, together with a valuation adapted to the non-reduced structure, which allows us to analyze generators and their degrees of freedom in a systematic way. In particular, each stratum is affine when u=1,2; and each stratum is isomorphic to an algebraic torus times an affine space when u=v,v-1,v-2. We consequently compute the Poincaré polynomials of the punctual Hilbert scheme of points on curves x^u y^v=0 when u=1,2,v-2,v-1,v.

What carries the argument

The weak diagonal partition, a combinatorial object that indexes strata obtained from filtrations on ideals and a valuation adapted to the non-reduced structure of the curve x^u y^v=0.

If this is right

  • The Poincaré polynomial of the punctual Hilbert scheme is explicitly computable for u=1 and u=2.
  • The Poincaré polynomial is explicitly computable when u equals v, v-1 or v-2.
  • The colored Oblomkov-Rasmussen-Shende conjecture holds for the Hopf link whenever u=1 and v is arbitrary.

Where Pith is reading between the lines

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

  • The same filtration-and-valuation method might apply to non-reduced curves with more complicated singularities.
  • The resulting polynomials could supply new tests for link-homology conjectures on other torus links.
  • One could verify the stratification by computing the Hilbert scheme directly for the smallest values of u and v.

Load-bearing premise

The filtrations on ideals and the valuation adapted to the non-reduced structure produce a well-defined stratification whose strata are affine or torus times affine for the stated ranges of u and v.

What would settle it

Direct computation of the Betti numbers or Poincaré polynomial of the Hilbert scheme for small explicit values such as u=1 and v=3, checked against the sum over weak diagonal partitions.

read the original abstract

We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, $x^uy^v=0$. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on introducing filtrations on ideals, together with a valuation adapted to the non-reduced structure, which allows us to analyze generators and their degrees of freedom in a systematic way. In particular, each stratum is affine when $u=1,2$; and each stratum is isomorphic to an algebraic torus times an affine space, $(\mathbb{C}^*)^{m_1} \times \mathbb{C}^{m_2}$, when $u=v,v-1,v-2$. We consequently compute the Poincar\'e polynomials of the punctual Hilbert scheme of points on curves $x^uy^v=0$ when $u=1,2,v-2,v-1,v$. As an application, we prove the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link for $u=1, v$ arbitrary, showing that the Poincar\'e polynomial is the row-colored link homology up to change of variables.

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.

Referee Report

2 major / 2 minor

Summary. The paper constructs a stratification of the punctual Hilbert scheme of points on the non-reduced nodal plane curve x^u y^v=0. Strata are indexed by a new combinatorial object called weak diagonal partitions, obtained via filtrations on ideals together with a valuation adapted to the non-reduced structure. For u=1,2 the strata are claimed to be affine spaces; for u=v, v-1, v-2 they are claimed to be algebraic tori times affine spaces. This yields explicit Poincaré polynomials for the listed ranges of u and v, and is applied to prove the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link when u=1 and v is arbitrary.

Significance. If the stratification is shown to be complete and free of hidden relations in the coordinate rings, the work would supply the first explicit Poincaré polynomials for punctual Hilbert schemes on these non-reduced curves and resolve a concrete case of the colored ORS conjecture. The systematic use of adapted filtrations and valuations offers a potentially reusable method for analyzing Hilbert schemes on other singular or non-reduced structures.

major comments (2)
  1. [Stratification construction (Section 3)] The load-bearing step is the assertion that the adapted valuation together with the filtrations produces a partition into strata whose coordinate rings are exactly those of affine spaces (u=1,2) or (C*)^{m1} × C^{m2} (u=v,v-1,v-2) with no additional equations arising from syzygies. The manuscript must supply an explicit verification that every ideal lies in one of the described strata and that the generators' degrees of freedom are precisely as counted; without this check the computed Poincaré polynomials and the proof of the colored ORS conjecture for the Hopf link do not follow.
  2. [Definition of weak diagonal partitions and valuation (Section 2)] The definition of weak diagonal partitions and the precise adaptation of the valuation to the non-reduced equation x^u y^v=0 are introduced without a self-contained check that the resulting stratification is exhaustive for the stated ranges of u and v. A concrete example for small u,v (e.g., u=1, v=3) showing that no ideal is missed or overcounted would make the claim verifiable.
minor comments (2)
  1. The abstract states the geometric types of the strata but does not indicate where in the text the isomorphism statements are proved; adding forward references to the relevant propositions would improve readability.
  2. Notation for the parameters m1 and m2 in the torus-affine description should be defined explicitly when first used rather than left implicit in the count of degrees of freedom.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the stratification. We address the two major comments below and will incorporate the requested checks and example into the revised manuscript.

read point-by-point responses

  1. Referee: [Stratification construction (Section 3)] The load-bearing step is the assertion that the adapted valuation together with the filtrations produces a partition into strata whose coordinate rings are exactly those of affine spaces (u=1,2) or (C*)^{m1} × C^{m2} (u=v,v-1,v-2) with no additional equations arising from syzygies. The manuscript must supply an explicit verification that every ideal lies in one of the described strata and that the generators' degrees of freedom are precisely as counted; without this check the computed Poincaré polynomials and the proof of the colored ORS conjecture for the Hopf link do not follow.

    Authors: We agree that a self-contained verification strengthens the argument. In the revision we will add a new subsection (3.4) that explicitly constructs the inverse map from each weak diagonal partition back to an ideal, shows that the filtration and adapted valuation recover the original generators without imposing extra syzygies, and confirms that the dimension count matches the claimed affine or torus-times-affine structure for the listed ranges of u and v. This will make the Poincaré polynomials and the Hopf-link application fully rigorous. revision: yes

  2. Referee: [Definition of weak diagonal partitions and valuation (Section 2)] The definition of weak diagonal partitions and the precise adaptation of the valuation to the non-reduced equation x^u y^v=0 are introduced without a self-contained check that the resulting stratification is exhaustive for the stated ranges of u and v. A concrete example for small u,v (e.g., u=1, v=3) showing that no ideal is missed or overcounted would make the claim verifiable.

    Authors: We will insert a fully worked example for u=1, v=3 immediately after the definition of weak diagonal partitions. The example enumerates all admissible partitions, constructs the corresponding ideals via the filtration, verifies that every monomial ideal in the punctual Hilbert scheme arises uniquely, and confirms there are no omissions or overlaps. This concrete check will demonstrate exhaustiveness for the base case and illustrate the general procedure. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses newly defined filtrations, valuation, and weak diagonal partitions

full rationale

The paper constructs a stratification of the punctual Hilbert scheme directly from original definitions: filtrations on ideals together with a valuation adapted to the non-reduced structure x^u y^v=0. Strata are indexed by the newly introduced combinatorial object 'weak diagonal partition'. For the stated ranges of u and v the strata are shown to be affine or (C*)^{m1} x C^{m2} by explicit analysis of generators and degrees of freedom. Poincaré polynomials are then obtained by summing over these strata, and the colored ORS conjecture for the Hopf link follows as a consequence. No step reduces by construction to a fitted parameter, self-cited uniqueness theorem, or renamed known result; the central claims rest on the independent verification that the stratification is complete and that the coordinate rings have the claimed form. This is a self-contained geometric construction with no load-bearing self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based on abstract only; the paper introduces a new combinatorial object and relies on domain-specific constructions whose precise axioms are not visible.

axioms (1)
  • domain assumption Existence of a valuation adapted to the non-reduced structure of the curve
    Invoked to analyze generators and degrees of freedom in the ideals.
invented entities (1)
  • weak diagonal partition no independent evidence
    purpose: Index the strata of the punctual Hilbert scheme
    New combinatorial object defined to label the strata

pith-pipeline@v0.9.0 · 5499 in / 1258 out tokens · 38817 ms · 2026-05-13T18:59:06.138723+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    GesselandR.P.Stanley,‘Algebraic Enumeration ’, in Handbook of Combinatorics Vol

    I.M. GesselandR.P.Stanley,‘Algebraic Enumeration ’, in Handbook of Combinatorics Vol. 2.1995, pp.1021–1061

    work page 1995

  2. [2]

    Oblomkov, J

    A. Oblomkov, J. Rasmussen, V. Shende.The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link. With an appendix by Eugene Gorsky.Geom. Topol. 22 (2018), no. 2, 645–691

    work page 2018

  3. [3]

    Maulik.Stable pairs and the HOMFLY polynomial.Invent

    D. Maulik.Stable pairs and the HOMFLY polynomial.Invent. Math. 204 (2016), no. 3, 787–831

    work page 2016

  4. [4]

    Cautis.Remarks on coloured triply graded link invariants.Algebr

    S. Cautis.Remarks on coloured triply graded link invariants.Algebr. Geom. Topol. 17.6 (2017), pp. 3811–3836

    work page 2017

  5. [5]

    Kivinen, E

    O. Kivinen, E. Gorskiy, J. Simental.Algebra and geometry of link homology.Bulletin of the London Mathematical Society 55 (2023), no. 2, 537-591

    work page 2023

  6. [6]

    Bejleri.The motivic Hilbert zeta function of a planar n-fold thickening of a smooth curve.https://math.umd

    D. Bejleri.The motivic Hilbert zeta function of a planar n-fold thickening of a smooth curve.https://math.umd. edu/~dbejleri/non_reduced_hilbert_zeta.pdf

    work page

  7. [7]

    Torus link homology

    M. Hogancamp, A. Mellit.Torus Link Homology.ArXiv:1909.00418[math.GT]

    work page arXiv 1909

  8. [8]

    Brion, E

    M. Brion, E. Peyre.The virtual Poincaré polynomials of homogeneous spaces.Compositio Mathematica 134, 319–335 (2002)

    work page 2002

  9. [9]

    Pol.Singularités libres, formes et résidus logarithmiques

    D. Pol.Singularités libres, formes et résidus logarithmiques. Mathématiques générales[math.GM]. Université d’Angers, 2016. Français. NNT : 2016ANGE0021 . tel-01441450

    work page 2016

  10. [10]

    Conners.Row-Column mirror symmetry for colored torus knot homology.Selecta Mathematica, 30 (2024), no

    L. Conners.Row-Column mirror symmetry for colored torus knot homology.Selecta Mathematica, 30 (2024), no. 97

    work page 2024

  11. [11]

    Ellingsrud, S.A

    G. Ellingsrud, S.A. Stromme.On the homology of the Hubert scheme of points in the plane.Inventiones math- ematicae 87 (1987): 343-352

    work page 1987

  12. [12]

    Lusztig, J

    G. Lusztig, J. Smelt.Fixed point varieties on the space of lattices.Bull. London Math. Soc. 23 (1991), no. 3, 213–218

    work page 1991

  13. [13]

    Gorsky, M

    E. Gorsky, M. Mazin, A. Oblomkov.Generic curves and non-coprime Catalans.Journal of Algebraic Geometry

    work page