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Ballico, Edoardo; Bernardi, Alessandra

Tensor ranks on tangent developable of Segre varieties. (English) Zbl 1282.14090

Linear Multilinear Algebra 61, No. 7, 881-894 (2013).
In the projective space that parametrizes tensors of given type (modulo scalar multiplication), the variety \(X\) of decomposable tensors corresponds to the Segre embedding of a product of lower-dimensional projective spaces. A general element in the first secant variety of \(X\) corresponds to a tensor of rank \(2\). However, the secant variety contains the tangent variety \(T(X)\) in its closure, and tensors in the tangent variety may have (and usually do have) larger rank. The aim of the authors is the study of the rank of tensors in \(T(X)\). They obtain a complete description, which is effectively computable, by looking at the Geometry of sets of points that evince the rank of a tensor \(t\in T(X)\). Similar results, but with a different approach, were obtained independently by J. Buczynski and J. M. Landsberg [Linear Algebra Appl. 438, No. 2, 668–689 (2013; Zbl 1268.15024)].
Reviewer: Luca Chiantini (Siena)

Cited in 18 Documents

MSC:

14N05 Projective techniques in algebraic geometry

Keywords:

secant varieties; tensor rank; tangent developable; Segre varieties; Comon’s conjecture

Citations:

Zbl 1268.15024
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Full Text: DOI arXiv

References:

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