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 $\lambda ⊢n+1$ 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 $J/{\mathfrak{m}}_{p}J$, 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 $16$.

No change to the mathematical content. 37 pages, fixed MathJax delimiters in the abstract