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arxiv: 2606.08120 · v2 · pith:MULRTID6new · submitted 2026-06-06 · 🧮 math.AG · math.AC· math.CO

Deformation Theory and Torus-Fixed Geometry of the Nested Hilbert Scheme of Points

Chenyang Zhao This is my paper

Pith reviewed 2026-06-27 19:03 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.CO
keywords nested Hilbert schemetorus fixed pointsdeformation theoryYoung diagramscompatibility kernelarm-leg weightsremovable corners
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The pith

The tangent space to the nested Hilbert scheme at a torus-fixed point is the kernel of a compatibility map that shortens standard arm-leg weights by a rule tied to the removable corner.

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

The paper studies the nested Hilbert scheme of points on the affine plane, consisting of pairs of ideals where one is contained in the other and their colengths differ by one. It recalls that first-order deformations of a single ideal I are given by Hom(I, R/I), then shows that for a nested pair the tangent space is the kernel of the natural map from the direct sum of two such Hom spaces into Hom(I, R/J). Torus-fixed points are labeled by a partition together with one of its removable corners, which also marks a one-dimensional socle generator. The central calculation produces an explicit weight formula for this tangent space by adjusting the usual arm and leg lengths according to a shortening rule determined by the corner.

Core claim

For a nested pair of ideals I subset J corresponding to a partition lambda with removable corner c, the tangent space is the kernel of Hom(I, R/I) plus Hom(J, R/J) to Hom(I, R/J). At the torus fixed point (I_lambda, I_{lambda without c}), the weights in the tangent space are obtained from the standard arm-leg weights by applying a shortening rule determined by c.

What carries the argument

The compatibility kernel of the map Hom(I,R/I) ⊕ Hom(J,R/J) → Hom(I,R/J), which defines the tangent space to the nested scheme and modifies the weights at fixed points.

If this is right

  • The blow-up map to the product of the ordinary Hilbert scheme and the plane has fibers that are projective spaces whose fixed points are addable boxes of the smaller diagram.
  • The universal family, the blow-up geometry, and Young diagram combinatorics together determine the local geometry of the nested scheme.
  • The tangent weight formula can be verified by computing the compatibility kernel from monomial syzygies for all partitions of size at most 16.

Where Pith is reading between the lines

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

  • The shortening rule may allow closed-form expressions for tangent-space Euler characteristics on larger nested Hilbert schemes.
  • The same kernel construction could be applied to nestings of higher length or on other toric surfaces.
  • Combining the weight formula with other torus actions might produce character formulas for the full nested Hilbert scheme.

Load-bearing premise

The tangent space to the nested Hilbert scheme is precisely the kernel of that compatibility map between the Hom spaces.

What would settle it

A direct computation of the tangent space dimension or weights for a specific partition like (3,1) that differs from the predicted shortened arm-leg weights would falsify the formula.

read the original abstract

In this paper, we study the nested Hilbert scheme $(\mathbb{A}^2)^{[n,n+1]}=\mathrm{Hilb}^{n,n+1}(\mathbb{A}^2)$ from a combination of deformation theory, torus actions, and Young diagram combinatorics. We first recall the scheme theory and functor basics needed to define Hilbert schemes. We then use a classic result on first-order deformations to identify $T_I(\mathbb{A}^2)^{[n]}\cong \mathrm{Hom}_{\mathbb{C}[x,y]}(I,\mathbb{C}[x,y]/I)$. For a nested pair $I\subset J$, with $\dim_{\mathbb{C}}\mathbb{C}[x,y]/I=n+1$ and $\dim_{\mathbb{C}}\mathbb{C}[x,y]/J=n$, the tangent space becomes a compatibility kernel $T_{(I,J)}(\mathbb{A}^2)^{[n,n+1]}\cong \ker(\mathrm{Hom}(I,R/I)\oplus \mathrm{Hom}(J,R/J)\to \mathrm{Hom}(I,R/J))$. The torus-fixed points are indexed by a partition $\lambda\vdash n+1$ together with a removable corner $c$ of its Young diagram. This corner is not only combinatorial, but also the monomial form of a one dimensional socle direction in $R/I_\lambda$. The blow-up map to $(\mathbb{A}^2)^{[n]}\times \mathbb{A}^2$ has fibres given by projective spaces of one-dimensional quotients of $J/\mathfrak m_pJ$, whose torus-fixed points are addable boxes of the smaller diagram. These two local fibres explain how the universal family, the blow-up geometry, and Young diagram combinatorics come together in the study of the local geometry of the nested Hilbert scheme of points. Finally, we derive the tangent weight formula at a fixed point $(I_\lambda,I_{\lambda\setminus c})$ in the torus convention used in the paper. Using the standard arrow basis, we show in the proof how the arm-leg weights are modified by the compatibility kernel through a shortening rule determined by $c$. A Macaulay2 verification computes the compatibility kernel from monomial syzygies and checks the weight formula for all partitions of size at most $16$.

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

0 major / 2 minor

Summary. The paper studies the nested Hilbert scheme Hilb^{n,n+1}(A^2) using deformation theory and torus actions. It recalls that the tangent space at a point I in the ordinary Hilbert scheme is Hom(I, R/I), extends this to nested pairs (I,J) with I ⊂ J via the compatibility kernel ker(Hom(I,R/I) ⊕ Hom(J,R/J) → Hom(I,R/J)), indexes the torus-fixed points by a partition λ ⊢ n+1 together with a removable corner c, and derives an explicit formula for the tangent weights at (I_λ, I_{λ∖c}) by modifying the standard arm-leg weights through a shortening rule induced by the kernel condition on the monomial generators corresponding to c. The formula is checked computationally via Macaulay2 for all partitions of size ≤16.

Significance. If the derivation holds, the explicit tangent-weight formula supplies a combinatorial description of the tangent spaces at torus-fixed points of the nested Hilbert scheme. This is a modest but concrete advance that combines standard homological identifications with Young-diagram combinatorics and supplies a Macaulay2 verification; such formulas are useful for subsequent calculations of equivariant cohomology or K-theory on these spaces.

minor comments (2)
  1. The precise torus-weight convention (positive vs. negative roots, normalization of the action on the coordinate ring) is referenced but not restated in a single location; adding a short table of weights for the standard basis monomials in a small example (e.g., λ=(2,1)) would improve readability.
  2. The statement that the corner c corresponds to a one-dimensional socle direction in R/I_λ is asserted without an explicit reference to the monomial basis; a one-sentence reminder of why the socle is one-dimensional at a removable box would help readers unfamiliar with the monomial ideal correspondence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the standard identification T_I ≅ Hom(I, R/I) for ordinary Hilbert schemes (a classic result in deformation theory) and extends it to the nested case by defining the tangent space as the kernel of the natural compatibility map Hom(I,R/I) ⊕ Hom(J,R/J) → Hom(I,R/J). The tangent-weight formula at torus-fixed points (I_λ, I_λ∖c) is then obtained by restricting the torus action to this kernel and applying the arrow basis to track how arm-leg weights are shortened by the removable corner c; this is a direct linear-algebraic computation on monomial generators. The argument is self-contained, uses no fitted parameters presented as predictions, invokes no load-bearing self-citations, and is cross-checked by explicit Macaulay2 computation of the kernel for all partitions of size ≤16. No step reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard results from deformation theory and scheme theory; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math T_I (A^2)^[n] ≅ Hom_{C[x,y]}(I, C[x,y]/I) from first-order deformations
    Classic result recalled to identify the tangent space.

pith-pipeline@v0.9.1-grok · 5960 in / 1274 out tokens · 17589 ms · 2026-06-27T19:03:56.896896+00:00 · methodology

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Reference graph

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