[Paper Review] On the symmetric subscheme of Hilbert scheme of points

Kyungyong Lee|arXiv (Cornell University)|Aug 26, 2007

Advanced Theoretical and Applied Studies in Material Sciences and Geometry6 references3 citations

TL;DR

This paper provides an explicit algebraic description of the most symmetric affine open subscheme $U$ of the Hilbert scheme $\text{Hilb}^{d+1}(\mathbb{C}^d)$ for $d \geq 3$, showing it is isomorphic to a quotient of a symmetric algebra over a Schur module. It proves that $\text{Hilb}^n(\mathbb{C}^d)$ is reducible for $n > d \geq 12$ by constructing a large-dimensional family of non-radical ideals, using Schur modules and the Littlewood-Richardson rule to analyze the Hilbert function and dimension of $U$. The key contribution is a geometric and representation-theoretic characterization of $U$ via Schur modules and ideal projectors.

ABSTRACT

We consider the Hilbert scheme Hilb^{d+1}(C^d) of (d+1) points in affine d-space C^d (d > 2), which includes the square of any maximal ideal. We describe equations for the most symmetric affine open subscheme of Hilb^{d+1}(C^d), in terms of Schur modules. In addition we prove that Hilb^{d+1}(C^d) is reducible for n>d>11.

Motivation & Objective

• To describe the equations of the most symmetric affine open subscheme $U$ of $\text{Hilb}^{d+1}(\mathbb{C}^d)$ for $d \geq 3$.
• To prove that $\text{Hilb}^n(\mathbb{C}^d)$ is reducible for $n > d \geq 12$ by constructing a large-dimensional family of non-radical ideals.
• To provide a representation-theoretic framework using Schur modules and the Littlewood-Richardson rule to analyze the Hilbert function of $U$.
• To conjecture a generalization of the description of $U$ to higher-degree monomial bases in $\mathbb{C}[\mathbf{x}]$.

Proposed method

• The symmetric affine subscheme $U$ is defined as the open subset of $\text{Hilb}^{d+1}(\mathbb{C}^d)$ where $\{1, x_1, \dots, x_d\}$ forms a $\mathbb{C}$-basis of $\mathbb{C}[\mathbf{x}]/I$, ensuring maximal symmetry.
• The coordinate ring of $U$ is described as $\mathbb{C}^d \times \text{Spec}\left(\frac{\text{Sym}^\bullet(\mathbb{S}_{(3,1,1,\dots,1,0)}V)}{\langle \mathbb{S}_{(4,3,2,\dots,2,1)}V \rangle}\right)$, where $V$ is a $d$-dimensional $\mathbb{C}$-vector space.
• An injective homomorphism $j: \mathbb{S}_{(4,3,2,\dots,2,1)}V \hookrightarrow \text{Sym}^2(\mathbb{S}_{(3,1,1,\dots,1,0)}V)$ is used to define the quotient ring structure.
• The Hilbert function $H(r)$ of the quotient ring is bounded below using the Littlewood-Richardson rule, showing that $H(r)$ grows faster than $\mathcal{O}(r^{k\binom{d-k}{2}})$ for any $k=0,\dots,d-1$, implying high dimensionality.
• A family of ideals is constructed via $d \times (d+1)$ matrices and ideal projectors, yielding a $193$-dimensional family for $d=13$, exceeding the dimension of the radical component.
• For $d=12$, a $156$-dimensional family is constructed, matching the dimension of the radical component but not contained in it, proving reducibility.

Experimental results

Research questions

• RQ1 What is the algebraic structure of the most symmetric affine open subscheme $U$ of $\text{Hilb}^{d+1}(\mathbb{C}^d)$ for $d \geq 3$?
• RQ2 Can the Hilbert scheme $\text{Hilb}^n(\mathbb{C}^d)$ be shown to be reducible for $n > d \geq 12$ using explicit constructions of non-radical ideals?
• RQ3 How does the Hilbert function of the coordinate ring of $U$ grow, and what does this imply about the dimension of $U$?
• RQ4 Can the equations defining $U$ be generalized to higher-degree monomial bases in $\mathbb{C}[\mathbf{x}]$?
• RQ5 What is the relationship between the ideal generated by $\mathbb{S}_{(4,3,2,\dots,2,1)}V$ and the principal (radical) component of $U$?

Key findings

• The symmetric affine subscheme $U \subset \text{Hilb}^{d+1}(\mathbb{C}^d)$ is isomorphic to $\mathbb{C}^d \times \text{Spec}\left(\frac{\text{Sym}^\bullet(\mathbb{S}_{(3,1,1,\dots,1,0)}V)}{\langle \mathbb{S}_{(4,3,2,\dots,2,1)}V \rangle}\right)$, providing an explicit algebraic description via Schur modules.
• For $d=13$, a $193$-dimensional family of ideals is constructed, exceeding the $182$-dimensional closure of the radical component, proving that $U$ is reducible for $d \geq 13$.
• For $d=12$, a $156$-dimensional family of ideals is constructed, matching the dimension of the radical component but not contained in it, proving reducibility for $d=12$.
• The Hilbert function $H(r)$ of the quotient ring satisfies $H(r) \geq \sum_{\lambda: \lambda_{d-k}+\cdots+\lambda_d \leq rk} m_\lambda \cdot \dim_{\mathbb{C}} \mathbb{S}_\lambda$ for $r \geq 2$, $k=0,\dots,d-1$, suggesting faster-than-polynomial growth.
• The dimension of $\mathbb{S}_{(3,1,1,\dots,1,0)}V$ is $d\binom{d+1}{2} - d$, and the dimension of $\mathbb{S}_{(4,3,2,\dots,2,1)}V$ is $\frac{d^2(d^2 - 4)}{3}$, which are used to compute $H(1)$ and $H(2)$ explicitly.
• The conjecture that the Hilbert polynomial $\tilde{H}(r)$ equals $H(r)$ for all $r \geq 0$ is verified for $d=3$, and holds for $d=4,5$ via Macaulay 2 computations.