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

Abstract

In this paper, we study the nested Hilbert scheme $(\mathbb{A}^2)^{[n,n+1]}=\mathrm{Hilb}^{n,n+1}(\mathbb{A}^2)$ 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 $16$.