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

Abstract: In this paper, we study the nested Hilbert scheme $$(\mathbb{A}<sup>2)<sup>{[n,n+1]}=\mathrm{Hilb}<sup>{n,n+1}(\mathbb{A}<sup>2)$$ 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 $$T_I(\mathbb{A}^{2)^{[n]}\cong} \mathrm{Hom}<em>{\mathbb{C}[x,y]}(I,\mathbb{C}[x,y]/I)$$. For a nested pair $I\subset J$, with $\dim</em>{\mathbb{C}}\mathbb{C}[x,y]/I=n+1$ 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 $λ\vdash n+1$ 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 $R/I_λ$. The blow-up map to $$(\mathbb{A}^{2)^{[n]}\times} \mathbb{A}<sup>2$$ 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 $(I_λ,I_{λ\setminus c})$ 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 $16$.

• The paper introduces a compatibility kernel method to compute tangent spaces for nested Hilbert schemes with rigorous computational verification.
• It combines torus-fixed analysis and Young diagram combinatorics to derive an explicit shortening rule for tangent weights.
• Computational experiments up to partition size 16 confirm the combinatorial tangent weight formula, supporting its implications for equivariant geometry.

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

Overview and Objectives

The paper "Deformation Theory and Torus-Fixed Geometry of the Nested Hilbert Scheme of Points" (2606.08120) provides a comprehensive and technical exposition of the nested Hilbert scheme of points $S^{[n,n+1]}$ on the affine plane $S = \mathbb{A}^2$, synthesizing deformation-theoretic formalism, torus actions, and Young diagram combinatorics. The analysis precisely characterizes tangent spaces at nested fixed points via compatibility kernels and presents a detailed account of how geometric, algebraic, and combinatorial structures interplay in the local geometry of $S^{[n,n+1]}$. Explicit computational verification up to partition size $16$ provides robust support for the combinatorial tangent weight formula central to equivariant geometry and subsequent enumerative applications.

Background: Hilbert Schemes and Deformation Theory

The Hilbert scheme $S^{[n]}$ parametrizes zero-dimensional closed subschemes of length $n$ in $\mathbb{A}^2$. Fogarty's theorem guarantees that $S^{[n]}$ is smooth and of dimension $2n$ when $S$ is a smooth surface [Fogarty1968]. The moduli-theoretic foundation is laid through the functor of points and deformation theory: first-order embedded deformations of subschemes are governed by the Hom space $S = \mathbb{A}^2$0, with $S = \mathbb{A}^2$1 an ideal of colength $S = \mathbb{A}^2$2 in the coordinate ring $S = \mathbb{A}^2$3. Zariski tangent spaces at closed points are canonically interpreted via morphisms from the spectrum of dual numbers, consistent with functorial formulations.

Nested Hilbert Scheme: Incidence Structure and Geometry

The nested Hilbert scheme $S = \mathbb{A}^2$4 parametrizes pairs of ideals $S = \mathbb{A}^2$5, with $S = \mathbb{A}^2$6 and $S = \mathbb{A}^2$7. Here, inclusion is reversed relative to subscheme containment, so $S = \mathbb{A}^2$8 defines the larger subscheme. The incidence condition, viewed as a closed embedding inside $S = \mathbb{A}^2$9, leads to a dimension count of $S^{[n,n+1]}$0, and Cheah's work establishes its smoothness [Cheah1998].

The principal maps emanating from $S^{[n,n+1]}$1 are:

• $S^{[n,n+1]}$2, the universal family over $S^{[n,n+1]}$3, associating $S^{[n,n+1]}$4 to $S^{[n,n+1]}$5, with $S^{[n,n+1]}$6 the residual point determined by the locus where $S^{[n,n+1]}$7 is supported.
• $S^{[n,n+1]}$8, realizing $S^{[n,n+1]}$9 as the blow-up of $16$0 along the universal family $16$1 [Lehn1999], [Ryan2020].

The difference in local geometry between these two maps is elucidated via socle fibers (removable corners) and generator directions (addable boxes), illustrating how scheme structure and infinitesimal data are encoded combinatorially.

Young Diagram Combinatorics and Monomial Ideals

Monomial ideals of finite colength in $16$2 are canonically in bijection with partitions $16$3, indexed by their Young diagrams $16$4. The combinatorial structure:

• Removable corners $16$5 correspond to socle directions of $16$6, reflecting one-dimensional subspaces killed by the maximal ideal.
• Addable boxes of the smaller diagram encode minimal generator directions in $16$7, with $16$8.

The arm and leg statistics ($16$9, $S^{[n]}$0) for $S^{[n]}$1 are determined and serve as essential data for tangent weight formulas.

Torus Action and Tangent Space Representations

The torus $S^{[n]}$2 acts on $S^{[n]}$3 by scaling coordinates. The fixed points of $S^{[n]}$4 are monomial ideals; for $S^{[n]}$5, the tangent representation decomposes into one-dimensional weight spaces, captured explicitly via the action on $S^{[n]}$6. The canonical arm-leg formula is: $S^{[n]}$7 with $S^{[n]}$8 the torus coordinates. This decomposition is compatible with established conventions [Nakajima1999], [Haiman1998], [EllingsrudStromme1987].

Tangent Spaces to Nested Hilbert Schemes: The Compatibility Kernel

A central technical result is the description of the tangent space at a nested pair $S^{[n]}$9 as a compatibility kernel: $n$0 where the map $n$1 imposes compatibility via $n$2, with $n$3 and $n$4.

At torus-fixed points, indexed by pairs $n$5 ($n$6 a removable corner of $n$7), the tangent representation is modified locally by the compatibility kernel, producing the so-called shortening rule for tangent weights. Specifically,

• For boxes left of $n$8, $n$9 is replaced by $\mathbb{A}^2$0.
• For boxes below $\mathbb{A}^2$1, $\mathbb{A}^2$2 is replaced by $\mathbb{A}^2$3.
• All other weights are unchanged.

The total number of weights is $\mathbb{A}^2$4, matching the dimension of $\mathbb{A}^2$5. This formulation extends the classical tangent character for $\mathbb{A}^2$6 and is consistent with enumerative and localization data [Cheah1998], [ChaputEvain2015], [KonckiZielenkiewicz2025].

Computational Verification and Numerical Results

A Macaulay2 script is provided and verifies the compatibility kernel formula against the combinatorial shortening rule for all partitions of size up to $\mathbb{A}^2$7. For each partition-removable corner pair, the algorithm computes the kernel of the syzygy-induced linear system and confirms agreement with the predicted weight multiset. This finite verification (2455 tests) demonstrates correctness and rigidity of the combinatorial tangent weight prescription.

Strong numerical claim: The computed weight multiset agrees precisely with the combinatorial shortening rule for every tested nested fixed point, with no discrepancies observed for $\mathbb{A}^2$8.

Implications and Future Directions

The deformation-theoretic framework provided here clarifies the connection between scheme-theoretic geometry, torus-equivariant structure, and combinatorial representation theory in the context of nested Hilbert schemes. The explicit compatibility kernel enables concrete computations of tangent spaces crucial for equivariant localization, intersection theory, and virtual fundamental class constructions (cf. [GholampourSheshmaniYau2020]). The proven correspondence between monomial socle directions (removable corners) and minimal generator directions (addable boxes) will facilitate further investigations into stratifications, cellular decompositions, and derived invariants.

Computational methods leveraging monomial ideals and syzygy criteria can be extended to broader classes of incidence-type moduli spaces, possibly incorporating higher-order deformations, curve-nesting, or more complex incidence schemes. The confirmed accuracy of combinatorial tangent weight formulas supports future applications in enumerative geometry, representation theory of rational Cherednik algebras, and the study of refined invariants (e.g., Macdonald polynomials, $\mathbb{A}^2$9-Catalan numbers).

Conclusion

The paper establishes an authoritative and self-contained presentation of the tangent geometry and deformation structure underlying the nested Hilbert scheme of points on the affine plane. The combination of deformation theory, torus action, and explicit Young diagram combinatorics yields a robust, algorithmically tractable framework for computing tangent spaces at torus-fixed points, with rigorous computational verification. This synthesis both clarifies fundamental geometric structures and enables further advances in the study of moduli spaces, equivariant invariants, and combinatorial algebraic geometry.