[Paper Review] Induction for secant varieties of Segre varieties

Hirotachi Abo, Giorgio Ottaviani+1 more|ArXiv.org|Jul 7, 2006

Tensor decomposition and applications13 references4 citations

TL;DR

This paper introduces an inductive method to compute the dimension of secant varieties of Segre varieties by reducing high-dimensional problems to lower-dimensional partial secant varieties through successive specializations and projections. The key contribution is a complete classification of defective $t$-secant varieties for $t \leq 6$, identifying rare defective cases such as $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n \times \mathbb{P}^n$ and $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^3$, and generalizing non-defectivity results for tensor powers of $\mathbb{P}^n$. The method enables systematic analysis of tensor rank and border rank via algebraic geometry and computational algebraic techniques.

ABSTRACT

This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective $t$-secant varieties to Segre varieties for t < 7. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of P^n. We determine the set of p for which unbalanced Segre varieties have defective p-secant varieties. In addition, we show that the Segre varieties P^1 x P^1 x P^n x P^n and P^2 x P ^3 x P^3 are deficient and completely describe the dimensions of their secant varieties. In the final section we propose a series of conjectures about defective Segre varieties.

Motivation & Objective

• To develop an inductive framework for computing the dimension of secant varieties of Segre varieties, reducing complex high-dimensional cases to simpler, low-dimensional partial secant varieties.
• To classify all defective $t$-secant varieties of Segre varieties for $t \leq 6$, identifying rare exceptional cases beyond unbalanced configurations.
• To generalize a theorem by Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of $\mathbb{P}^n$, particularly for odd $n$.
• To determine the set of $p$ for which unbalanced Segre varieties have defective $p$-secant varieties, and to fully describe the dimensions of secant varieties for known defective cases like $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n \times \mathbb{P}^n$ and $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^3$.
• To propose and support a series of conjectures on the existence and classification of defective Segre varieties, based on computational evidence and inductive reduction.

Proposed method

• The inductive procedure uses successive specializations of points and projections to reduce the computation of $\dim \sigma_p(\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_k})$ to partial secant varieties in lower dimensions.
• The method relies on the concept of joins of projective varieties and the Zariski closure of linear spans of $p$-tuples of points from each factor variety.
• Key tools include Terracini’s Lemma and Monte Carlo techniques to verify non-defectivity in high-dimensional cases.
• The authors use tensor algebra and hypermatrix decompositions to interpret secant varieties as spaces of tensors of border rank at most $p$, linking geometric properties to algebraic invariants.
• The reduction process is formalized via recursive reduction lemmas (e.g., Corollary 3.9 and Theorem 3.4) that simplify complex configurations into known base cases.
• Computational algebra systems like CoCoA, Macaulay2, and Singular are used to support and verify results, especially for testing perfectness and non-defectivity.

Experimental results

Research questions

• RQ1 For which values of $p$ do unbalanced Segre varieties $\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_k}$ have defective $p$-secant varieties?
• RQ2 Are all defective $\sigma_s(X)$ for balanced Segre varieties of the form $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2} \times \mathbb{P}^{n_3}$ classified into four known families: unbalanced, $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n \times \mathbb{P}^n$, $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^3$, and $\mathbb{P}^2 \times \mathbb{P}^n \times \mathbb{P}^n$ with $n$ even?
• RQ3 Is every Segre variety of the form $({\mathbb{P}^n})^k$ non-defective for $k \geq 3$, except possibly $({\mathbb{P}^2})^3$ and $({\mathbb{P}^1})^4$?
• RQ4 For balanced, numerically perfect Segre varieties with odd $n_1, n_2, n_3$, is $X$ always perfect (i.e., non-defective)?
• RQ5 Does every defective Segre variety fall into one of four classes: unbalanced, $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n \times \mathbb{P}^n$, $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^3$, or $\mathbb{P}^2 \times \mathbb{P}^n \times \mathbb{P}^n$ with $n$ even?

Key findings

• The paper provides a complete classification of defective $t$-secant varieties for $t \leq 6$, showing that only a few exceptional cases—such as $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n \times \mathbb{P}^n$ and $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^3$—are defective beyond unbalanced configurations.
• The dimensions of secant varieties for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n \times \mathbb{P}^n$ and $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^3$ are fully described, with explicit formulas derived via the inductive method.
• For odd $n$, the Segre variety $\mathbb{P}^k \times (\mathbb{P}^n)^{k+1}$ is proven to be perfect, generalizing non-defectivity results for tensor powers of $\mathbb{P}^n$.
• The method reduces complex tensor rank problems to simpler base cases, and the authors verify that all balanced, numerically perfect 3-factor Segre varieties with $n_i \leq 30$ are perfect using Monte Carlo techniques.
• The paper shows that $T(2^3,0,1;5;0^3,10,2)$ is true by reducing it via Corollary 3.9 and Theorem 3.4 to known true statements about 3-secant varieties, demonstrating the inductive method’s effectiveness.
• The authors conjecture that all defective Segre varieties fall into exactly four classes: unbalanced, $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n \times \mathbb{P}^n$, $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^3$, and $\mathbb{P}^2 \times \mathbb{P}^n \times \mathbb{P}^n$ with $n$ even, based on extensive computational evidence.