ON THE CONCEPT OF k-SECANT ORDER OF A VARIETY

Chiantini, Luca; Ciliberto, Ciro

doi:10.1112/s0024610706022630

Cited by 80 publications

(114 citation statements)

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“…The above proof also recovers the (previously known) fact that general points in P 3 , namely those outside of the tangential variety of C, admit precisely two decompositions as linear combinations of two points in C. This holds more generally for elliptic normal curves of even degree, see [14,Proposition 5.2].…”

Section: Remark 62supporting

confidence: 74%

On the locus of points of high rank

Buczyński

Han

Mella

et al. 2017

European Journal of Mathematics

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Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X . Let W k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any GL n orbit of a homogeneous form.

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Section: Remark 62supporting

confidence: 74%

On the locus of points of high rank

Buczyński

Han

Mella

et al. 2017

European Journal of Mathematics

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“…Section 2.4). In fact, it has been proved that symmetric tensors of any order d ≥ 3 and of dimension 2 or 3 always have an essentially unique decomposition with probability 1 if their rank is strictly smaller than the generic rank [54]. Our conjecture is that this also holds true for non-symmetric tensors: all tensors with a sub-generic rank have a unique decomposition with probability 1, up to scale and permutation indeterminacies.…”

Section: Number Of Decompositionsmentioning

confidence: 74%

Tensor decompositions, alternating least squares and other tales

Comon

Luciani

Almeida

2009

Journal of Chemometrics

387

350

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aThis work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple 'bottlenecks', and on 'swamps'. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the enhanced line search (ELS) enhancement in these algorithms is discussed. Computer simulations feed this discussion.

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“…. , X k are projective varieties in P r , then Terracini's Lemma describes the tangent space to their join at a general point of it (see [25] or, for modern versions, [1], [5], [8], [10], [27]). …”

Section: Secant Varieties and Contact Locimentioning

confidence: 99%