Grassmann cactus variety and socle dimension

1. Problem Statement and Context

The paper addresses the characterization of Grassmann cactus varieties, denoted as $K_{r,k}(X)$. These varieties serve as a common generalization of two well-established concepts in algebraic geometry:

• Grassmann Secant Varieties: Defined by linear spans of smooth schemes (unions of $r$ distinct points).
• Cactus Varieties: Defined by linear spans of finite schemes of degree $\le r$ (allowing non-reduced structures).

The Core Problem:
In the definition of $K_{r,k}(X)$, one considers $k$-dimensional linear subspaces $E \subset V$ contained in the linear span $\langle R \rangle$ of a zero-dimensional subscheme $R \subset X$ with $\deg(R) \le r$. The set of all such $E$ forms the variety.
The authors investigate whether the class of subschemes $R$ required to generate these varieties can be restricted. Specifically, they ask: Is it necessary to consider all finite schemes of degree $r$, or can the definition be restricted to schemes with specific structural properties (specifically, low socle dimension) without changing the resulting variety?

2. Mathematical Definitions and Preliminaries

• Grassmann Cactus Variety:
  $$K_{r,k}(X) := \overline{\{E \in \text{Gr}(k, V) \mid \exists R \subset X, \dim R = 0, \deg R \le r, E \subset \langle R \rangle\}}$$
• Socle Dimension:
  For a finite local ring $A$ with maximal ideal $\mathfrak{m}$, the socle is $s = \text{Ann}_A(\mathfrak{m}) = \{a \in A \mid a \cdot \mathfrak{m} = 0\}$.
  A finite scheme $R$ has socle dimension at most $l$ ($\text{socdim}(R) \le l$) if, for every local component $A_i$ of $R$, $\dim_k(s_i) \le l$.
  • Note: A scheme is Gorenstein if and only if $\text{socdim}(R) \le 1$.
• Hilbert Scheme of Subschemes: The paper utilizes $\text{Hilb}^k_R$, the Hilbert scheme of finite subschemes of degree $k$ within a fixed finite scheme $R$.

3. Methodology

The authors employ a reduction strategy combined with a detailed analysis of the geometry of subschemes within finite schemes.

1. Reduction to Local Case: The proof reduces the general problem to the case where $X$ is a single finite scheme $R$ of degree $r$.
2. Analysis of Subschemes (Lemmas 2 & 4):
   • They analyze the structure of subschemes of degree $r-1$ within a local scheme $R = \text{Spec}(A)$.
   • Lemma 2: Establishes that the reduced Hilbert scheme of subschemes of degree $r-1$ in a local scheme $R$ is isomorphic to the projective space of its socle, $\mathbb{P}(s)$. This links the geometry of subschemes directly to the socle dimension.
   • Fun Fact 3: Highlights that while the reduced structure is a projective space, the full Hilbert scheme $\text{Hilb}^{r-1}_R$ is not necessarily reduced (e.g., for $R=\text{Spec}(k[t]/(t^2))$).
3. Spanning Argument (Lemma 6):
   • The core technical step involves proving that if a finite scheme $R$ has a socle dimension $l > k$, its linear span $\langle R \rangle$ is already covered by the linear spans of its subschemes of degree $r-1$.
   • Specifically, if $\text{socdim}(R) > k$, then $\text{Gr}(k, \langle R \rangle) = \bigcup_{R' \subsetneq R} \text{Gr}(k, \langle R' \rangle)$.
   • The proof utilizes dimension counting on the Grassmannian and the parameter space of subschemes ($\mathbb{P}(s)$) to show that any $k$-plane in $\langle R \rangle$ lies in the span of a smaller subscheme.

4. Key Contributions and Results

Main Theorem (Theorem 1)

The central result of the paper is the characterization of Grassmann cactus varieties using schemes with bounded socle dimension:
$$K_{r,k}(X) = \overline{\{E \in \text{Gr}(k, V) \mid \exists R \subset X, \dim R = 0, \deg R \le r, \text{socdim}(R) \le k, E \subset \langle R \rangle\}}$$

Implication: To define the Grassmann cactus variety $K_{r,k}(X)$, it is sufficient to consider only finite subschemes $R$ whose local components have a socle dimension of at most $k$. One does not need to consider schemes with higher socle dimensions.

Corollary for $k=1$

The theorem recovers the known result for standard cactus varieties ($k=1$), where it suffices to use Gorenstein schemes ($\text{socdim} \le 1$). This confirms that the generalization to arbitrary $k$ is consistent with existing literature.

Optimality and Conjectures (Section 4)

The authors discuss whether the bound $\text{socdim} \le k$ is optimal.

• Conjecture 9: They conjecture that for a "sufficiently ample" embedding of a projective variety $X$, the irreducible components of $K_{r,k}(X)$ are in bijection with the irreducible components of the Hilbert scheme $\text{Hilb}^r(X)$ consisting of schemes with $\text{socdim} \le k$.
• If true, this implies the bound in Theorem 1 is sharp and cannot be lowered (e.g., one cannot generally define $K_{r,2}$ using only Gorenstein schemes).

5. Significance and Impact

1. Simplification of Calculations: By restricting the search space to schemes with low socle dimension, the complexity of calculating examples of Grassmann cactus varieties is significantly reduced. This is crucial for computational algebraic geometry.
2. Connection to Border Rank: The authors expect that exploiting $K_{r,k}(X)$ will lead to stronger "cactus barriers" for determinantal methods used in computing border rank, surpassing previous bounds (e.g., those by [EGOW18] or [GMOW19]).
3. Structural Insight: The paper provides a deep structural link between the socle dimension of an algebra and the linear span of the corresponding scheme. It demonstrates that high socle dimension (which implies "pathological" behavior in the scheme) does not contribute new points to the Grassmann cactus variety beyond what is already generated by lower socle dimension subschemes.
4. Future Research Direction: The work motivates the study of parameter spaces for schemes with low socle dimensions (referencing work in preparation [BB25], [Jag25], and [GKŠ25]). It opens questions about the reducibility of Hilbert schemes of schemes with $\text{socdim} \le 2$ (conjectured to be reducible for $r \ge 13$ in characteristic 0).

Summary

This paper establishes that the geometric complexity of Grassmann cactus varieties is governed by the socle dimension of the underlying finite schemes. The authors prove that restricting the definition to schemes with socle dimension $\le k$ yields the same variety as the unrestricted definition. This result unifies the theory of secant and cactus varieties, simplifies computational approaches to border rank problems, and sets the stage for further investigation into the geometry of Hilbert schemes with bounded socle dimensions.