A One-Variable Frame Construction For Irrational Components of Hilbert Schemes of Points

Ruoyu Wu

REVIEW 1 major objections 1 minor 5 references

A One-Variable Frame Construction For Irrational Components of Hilbert Schemes of Points

T0 review · 1 major / 1 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Over fields of characteristic zero, Hilb(A^n) has non-rational irreducible components for all n at least 10.

desk verdict 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. read the letter →

arxiv 2606.30386 v1 pith:FECYXUOJ submitted 2026-06-29 math.AG math.AC

Ruoyu Wu This is my paper · ORCID

classification math.AGmath.AC

keywords Hilbertschemesofpointsnon-rationalcomponentslocalcohomologyHartshorne-Raomoduleg^3_9curveframeconstructionaffinespace

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the 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.

That 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.

The 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.

This 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.

Referee Report

1 major / 1 minor

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.

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 (1)

[local-cohomology replacement argument (Introduction and main construction)] 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.

minor comments (1)

[Abstract] 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.

Simulated Author's Rebuttal

1 responses · 0 unresolved

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.

read point-by-point responses

Referee: 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.

Authors: 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: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; extension relies on external citations and independent local-cohomology argument

full rationale

The derivation begins from the cited Farkas-Pandharipande-Sammartano construction (n≥12) and Jelisiejew TNT frame, then substitutes a local-cohomology argument that invokes only the vanishing of low-degree Hartshorne-Rao modules for the complete g³₉ curve. No equation or step reduces the target statement (non-rational components for n≥10) to a quantity defined in terms of itself, to a fitted parameter renamed as prediction, or to a load-bearing self-citation chain. All load-bearing inputs are external to the present paper and are not shown to be equivalent to the conclusion by construction.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted beyond the stated reliance on vanishing of a Hartshorne-Rao module.

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Cite this review

Pith. "Pith review of A One-Variable Frame Construction For Irrational Components of Hilbert Schemes of Points." pith.science (2026). https://pith.science/paper/FECYXUOJ

@misc{pith2026260630386,
  author       = {Pith},
  title        = {Pith review of: A One-Variable Frame Construction For Irrational Components of Hilbert Schemes of Points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FECYXUOJ}},
  note         = {Machine review of arXiv:2606.30386}
}

abstract

Farkas, Pandharipande, and Sammartano constructed non-rational irreducible components of Hilbert schemes of points in affine space $\mathbb{A}^n$ for all $n \geq 12$. Their construction starts from Hilbert schemes of curves in $\mathbb{P}^3$, adjoins two auxiliary variables in order to apply Jelisiejew's TNT frame construction, and then doubles the number of variables. We give a one-variable variant of the construction. The new input is a local-cohomology replacement for the depth-three step in Jelisiejew's negative tangent computation. It uses the vanishing of the low-degree Hartshorne--Rao module for the complete $g^3_9$ curve source. As a consequence, over a field of characteristic zero, $\operatorname{Hilb}(\mathbb{A}^n)$ has non-rational irreducible components for all $n \geq 10$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 3 canonical work pages

[1]

Eisenbud and J

D. Eisenbud and J. Harris,The Kodaira dimension of the moduli space of curves of genus at least 23, Invent. Math. 90 (1987), 359–387

1987
[2]

Farkas, D

G. Farkas, D. Jensen, and S. Payne,The Kodaira dimension ofM 22 andM 23, arXiv:2005.00622. 10 RUOYU WU

work page arXiv 2005
[3]

Farkas, R

G. Farkas, R. Pandharipande, and A. Sammartano,Irrational components of the Hilbert scheme of points, arXiv:2405.11997

work page arXiv
[4]

Harris and D

J. Harris and D. Mumford,On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23–88

1982
[5]

Jelisiejew,Pathologies on the Hilbert scheme of points, arXiv:1812.08531

J. Jelisiejew,Pathologies on the Hilbert scheme of points, arXiv:1812.08531

work page arXiv

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.

[1] [1]

Eisenbud and J

D. Eisenbud and J. Harris,The Kodaira dimension of the moduli space of curves of genus at least 23, Invent. Math. 90 (1987), 359–387

1987

[2] [2]

Farkas, D

G. Farkas, D. Jensen, and S. Payne,The Kodaira dimension ofM 22 andM 23, arXiv:2005.00622. 10 RUOYU WU

work page arXiv 2005

[3] [3]

Farkas, R

G. Farkas, R. Pandharipande, and A. Sammartano,Irrational components of the Hilbert scheme of points, arXiv:2405.11997

work page arXiv

[4] [4]

Harris and D

J. Harris and D. Mumford,On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23–88

1982

[5] [5]

Jelisiejew,Pathologies on the Hilbert scheme of points, arXiv:1812.08531

J. Jelisiejew,Pathologies on the Hilbert scheme of points, arXiv:1812.08531

work page arXiv