[Paper Review] On the ideals of secant varieties of Segre varieties

J. M. Landsberg, L. Manivel|ArXiv.org|Nov 21, 2003

Tensor decomposition and applications13 references4 citations

TL;DR

This paper develops techniques for computing the ideals of secant varieties of Segre varieties, resolving a conjecture on generators of the first secant variety for three-factor Segre products and proving that the third-degree components cut out the second secant variety set-theoretically in all cases and ideal-theoretically when there are three factors. It provides a deterministic algorithm for finding ideal generators and shows that the ideal of the sixth secant variety of $\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3$ has no equations in degrees less than nine.

ABSTRACT

We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.

Motivation & Objective

• To establish general techniques for studying the ideals of secant varieties of rational homogeneous varieties, particularly Segre products.
• To resolve a conjecture by Garcia, Stillman, and Sturmfels on the generators of the first secant variety of three-factor Segre products.
• To determine low-degree components of the ideals of small-dimensional secant varieties of Segre products.
• To provide a deterministic algorithm for computing generators of the ideals of secant varieties of Segre varieties.
• To analyze the absence of low-degree equations in the ideal of $\sigma_6(\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3)$, relevant to matrix multiplication complexity.

Proposed method

• Leverages Schur duality and representation theory to analyze the structure of ideals in symmetric and exterior powers of tensor products.
• Applies Terracini’s lemma and third fundamental form calculations to determine the expected dimension of secant varieties.
• Uses a graph-theoretic and combinatorial approach to analyze the vanishing of multilinear forms in the ideal of secant varieties.
• Employs symmetrization and skew-symmetrization techniques to identify invariants that vanish on secant varieties.
• Develops a deterministic algorithm to compute generators of the ideals of secant varieties of Segre products.
• Validates results via computational checks using C++ code for specific cases, particularly in degree 12 for $\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3$.

Experimental results

Research questions

• RQ1 What are the generators of the ideal of the first secant variety of a three-factor Segre product, and do they cut out the variety set-theoretically or ideal-theoretically?
• RQ2 Can the third-degree components of the ideal of the second secant variety of a Segre product be shown to define the variety set-theoretically in general and ideal-theoretically in the three-factor case?
• RQ3 Are there equations of degree less than nine in the ideal of $\sigma_6(\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3)$?
• RQ4 What is the role of Schur duality and representation-theoretic decomposition in characterizing the ideals of secant varieties of Segre products?
• RQ5 Which specific symmetric tensors in $S^{12}(A \otimes B \otimes C)$ vanish on $\sigma_6(\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3)$?

Key findings

• The third-degree components $I_3(\sigma_2(X))$ cut out $\sigma_2(X)$ set-theoretically for any Segre product and ideal-theoretically when there are exactly three factors.
• The conjecture of Garcia, Stillman, and Sturmfels on the generators of the first secant variety is resolved affirmatively for three-factor Segre products.
• There are no equations in the ideal of $\sigma_6(\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3)$ in degrees less than nine.
• The module $S_{3333}A \otimes S_{3333}B \otimes S_{3333}C$ does not lie in the ideal of $\sigma_6(\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3)$, indicating that degree 12 forms are not sufficient to define the variety in low degrees.
• The paper provides a deterministic algorithm to compute generators of the ideals of secant varieties of Segre products, enabling systematic computation in low degrees.
• Computational verification using C++ code confirms that no low-multiplicity terms in degree nine vanish on $\sigma_6(\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3)$, supporting the absence of equations below degree nine.