[Paper Review] Linearly dependent and concise subsets of a Segre variety depending on k factors

Edoardo Ballico|arXiv (Cornell University)|Feb 22, 2020

Tensor decomposition and applications4 references4 citations

TL;DR

This paper investigates linearly dependent and concise subsets of Segre varieties depending on $k$ factors, focusing on circuits (equally dependent sets with $e(S) = 1$) and higher-dependency configurations. It establishes an upper bound of $\binom{s}{2} + s$ on the width $w(S)$ for circuits of size $s$, and fully classifies concise, equally dependent sets of size 6, showing they are either in specific examples with arbitrarily large width or have width at most 4, with $Y \cong (\mathbb{P}^1)^4$ if width is 4.

ABSTRACT

We study linearly dependent subsets with prescribed cardinality, $s$, of a multiprojective space. If the set $S$ is a circuit, we give an upper bound on the number of factors of the minimal multiprojective space containing $S$, while if $S$ has higher dependency this may be not true without strong assumptions. We describe the dependent subsets $S$ with $\#S=6$.

Motivation & Objective

• To understand the structure of linearly dependent and concise subsets in Segre varieties with respect to their width and dependency order.
• To determine whether upper bounds on width $w(S)$ hold for equally dependent sets beyond circuits ($e(S) = 1$).
• To classify all concise, equally dependent subsets $S \subset Y$ of size 6 in a multiprojective space $Y = \mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_k}$.
• To characterize the minimal multiprojective spaces containing such sets, especially when $w(S) = 4$ or $w(S)$ is arbitrarily large.

Proposed method

• Define $e(S) = h^1(\mathcal{I}_S(1,\dots,1))$ to measure linear dependency, with $e(S) = 1$ corresponding to circuits.
• Use Grassmann’s formula and properties of linear spans to relate $e(S)$ to the codimension of $\langle \nu(S) \rangle$ in $\mathbb{P}^r$.
• Apply the concept of concision to ensure that $w(S)$ equals the number of non-trivial factors in the minimal multiprojective space containing $S$.
• Analyze decompositions $S = A \cup B$ with $\#A = \#B = 3$, using irredundant spanning and intersection properties of linear spans.
• Use cohomological invariants and base field invariance to reduce to algebraic closure, preserving dimension and dependency data.
• Classify configurations via case analysis on intersections $E \cap A$, $E \cap B$, and widths of associated subspaces $Y'$, $Y''$, leading to width bounds and structural characterizations.

Experimental results

Research questions

• RQ1 For a circuit $S$ of size $s$, what is the maximum possible width $w(S)$, and can it be bounded in terms of $s$?
• RQ2 Does the width bound $w(S) \leq \binom{s}{2} + s$ extend to equally dependent sets with $e(S) > 1$?
• RQ3 What are the structural constraints on a concise, equally dependent set $S$ of size 6 in a Segre variety?
• RQ4 Can such sets have arbitrarily large width, and if so, under what geometric conditions?
• RQ5 When does $w(S) = 4$ imply $Y \cong (\mathbb{P}^1)^4$ for a concise, equally dependent $S$ of size 6?

Key findings

• The paper proves that for any circuit $S$ (i.e., $e(S) = 1$) of size $s$, the width satisfies $w(S) \leq \binom{s}{2} + s$, with equality possible for $s \geq 6$.
• For $e(S) > 1$, the bound fails without strong assumptions; the paper constructs examples with $e(S) > 1$ and arbitrarily large $w(S)$, showing the bound does not generalize.
• All concise, equally dependent sets $S$ of size 6 with $e(S) \geq 2$ are either as in Examples 4.1 or 4.2 (which have arbitrarily large width) or satisfy $w(S) \leq 4$.
• If $w(S) = 4$, then $Y \cong (\mathbb{P}^1)^4$, and this case occurs only when the minimal multiprojective space is the 4-fold product of $\mathbb{P}^1$.
• When $w(S) \leq 3$, the minimal space is $Y \cong \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ if $w(S) = 3$, and $w(S) = 2$ otherwise.
• The classification shows that no new configurations arise beyond the known examples and the $Y \cong (\mathbb{P}^1)^4$ case, under the given constraints.