[Paper Review] On the third secant variety

Jarosław Buczyński, J. M. Landsberg|arXiv (Cornell University)|Nov 29, 2011

Tensor decomposition and applications4 citations

TL;DR

This paper provides a complete classification of normal forms and ranks for tensors of border rank at most three, particularly focusing on the third secant variety of Segre embeddings. It establishes that the singular locus of σ₃(Seg(ℙA×ℙB×ℙC)) has codimension at least two, and identifies four distinct types of points on limiting trisecant planes for cominuscule varieties, with explicit geometric and algebraic characterizations of each type.

ABSTRACT

We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In particular there are at most four types of points on limiting trisecant planes for cominuscule varieties such as Grassmannians. We also show the singular locus of the first two secant varieties of all triple Segre products has codimension at least two.

Motivation & Objective

• To classify normal forms and ranks of tensors of border rank at most three, especially in the context of the third secant variety of Segre varieties.
• To analyze the geometry of limits of secant planes in a general setting, particularly for cominuscule varieties such as Grassmannians.
• To determine the structure and codimension of the singular locus of σ₃(Seg(ℙA×ℙB×ℙC)) for tensor spaces of small dimensions.
• To provide a differential-geometric analysis of trisecant planes and their limits, extending results on secant varieties and tangent varieties.
• To resolve open questions about the possible ranks of tensors of small border rank, which are poorly understood despite adequate tests for border rank.

Proposed method

• Uses normal form classification for points in σ₃(X)∖σ₂(X) on Segre varieties, distinguishing four types: (i) sum of three rank-one tensors, (ii) sum of a tangent vector and a rank-one tensor, (iii) sum of two tangent vectors along a curve, and (iv) sum of tangent vectors at two distinct points on a line in X.
• Applies differential-geometric techniques to study limits of secant planes, particularly analyzing the behavior of tangent vectors and their limits in the context of secant varieties.
• Employs the tangent map of a parameterization φ: W → ℙ(A⊗B⊗C) to compute the image of the differential at a point, showing it has full rank and thus proving smoothness at general points.
• Analyzes the dimension of the singular locus of σ₃(Seg(ℙA×ℙB×ℙC)) via explicit computation of the image of the tangent map and linear dependencies among generators.
• Uses 2×2 minors and slice matrices to reconstruct the triple of vectors (a₁,a₂,a₃), (b₁,b₂,b₃), (c₁,c₂,c₃) up to order and scaling, showing the preimage has six components.
• Constructs a table (Table 2) summarizing dimensions of σ₃ and upper bounds on the dimension of the singular locus for various tensor spaces, based on theoretical and computational results.

Experimental results

Research questions

• RQ1 What are the possible normal forms and ranks of tensors of border rank at most three, particularly in the third secant variety of a Segre embedding?
• RQ2 How do the limits of trisecant planes behave for cominuscule varieties such as Grassmannians, and how many distinct types of points arise on such limiting planes?
• RQ3 What is the codimension of the singular locus of σ₃(Seg(ℙA×ℙB×ℙC)) for various dimensions of A, B, and C?
• RQ4 Can the tangential variety of a Segre product be used to characterize smooth points of the second secant variety, and what does this imply about the singular locus?
• RQ5 How do the different types of points in σ₃(X)∖σ₂(X) relate topologically and geometrically—specifically, what are their codimensions and closure relations?

Key findings

• The singular locus of σ₃(Seg(ℙA×ℙB×ℙC)) has codimension at least two for all dimensions of A, B, and C, with codimension exactly two if one factor is ℂ³ and the others have dimension at least 3.
• There are exactly four types of points in σ₃(X)∖σ₂(X) for X = Seg(ℙA₁×⋯×ℙAₙ), with types (i)–(iv) corresponding to distinct geometric configurations of rank-one and tangent vectors.
• Points of type (i), i.e., sums of three rank-one tensors, contain a Zariski open subset of σ₃(X)∖σ₂(X), and are general in this variety.
• Points of type (ii) have codimension one in σ₃(X), those of type (iii) have codimension two, and those of type (iv) have codimension four, with type (iv) being in the closure of type (iii).
• For n=2, all points in σ₃(Seg(ℙA₁×ℙA₂))∖σ₂(Seg(ℙA₁×ℙA₂)) are of type (i), indicating a special case where higher-order secant structure collapses.
• The preimage of a general point in σ₃(Seg(ℙA×ℙB×ℙC)) under the parameterization map φ has six components, each isomorphic to (ℂ*)⁷, showing a 6:1 covering with discrete monodromy.