Generic forms of low Chow rank

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

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“…It is conjectured that varieties of split forms of degree d 3 are never defective. New cases have been recently proven in [147,148].…”

Section: Problem 5 Given a Homogeneous Polynomial F ∈ S D Find Thementioning

confidence: 99%

“…. , 1) d > 2 3(s − 1) < n [74] non-defective any any 2 [146] non-defective, except for cases above some numerical constraints [146,148] any any any 1 [149] non-defective, except for cases above 2 any any 2 n 3 any any 4.5. Varieties of Reducible Forms.…”

Section: Example 418 If P = [L D1mentioning

confidence: 99%

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

et al. 2018

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“…; see Arrondo andBernardi (2011), Shin (2012), Abo (2014), Torrance (2017), Catalisano et al (2015); (b) decompositions of the form f = Catalisano et al (2002), Ballico (2005), Abo and Vannieuwenhoven (2018); (c) decompositions of a more general form f = r i=1 Catalisano et al (2017).…”

Section: Generic K-rankmentioning

confidence: 99%

Algebraic Stories from One and from the Other Pockets

et al. 2018

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“…Shin then found a connection with Hilbert functions of unions of linear star-configurations Table 1. Known defective cases [17,37] and was able to use this to complete the n = 2 case for d ≤ 5 [33]. Abo [1] then adapted a technique of Brambilla and Ottaviani [10] to complete the n = 2 case and make significant progress towards the n = 3 and d = 3 cases.…”

Section: Introductionmentioning

confidence: 99%

All secant varieties of the Chow variety are nondefective for cubics and quaternary forms

Torrance¹,

Vannieuwenhoven²

2021

Trans. Amer. Math. Soc.

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The Chow rank of a form is the length of its smallest decomposition into a sum of products of linear forms. For a generic form, this corresponds to finding the smallest secant variety of the Chow variety which fills the ambient space. We determine the Chow rank of generic cubics and quaternary forms by proving nondefectivity of all involved secant varieties. The main new ingredient in our proof is the generalization of a technique by [Brambilla and Ottaviani, On the Alexander-Hirschowitz theorem, J. Pure Appl. Algebra, 2008] that consists of employing Terracini's lemma and Newton's backward difference formula to compute the dimensions of secant varieties of arbitrary projective varieties. Via this inductive construction, the proof of nondefectivity ultimately reduces to proving a number of base cases. These are settled via a computer-assisted proof because of the large dimensions of the spaces involved. The largest base case required in our proof consisted of computing the dimension of a vector space constructed from the 400th secant variety of a degree-82 Chow variety embedded in P 98769 .

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Generic forms of low Chow rank

Cited by 9 publications

References 11 publications

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Algebraic Stories from One and from the Other Pockets

All secant varieties of the Chow variety are nondefective for cubics and quaternary forms

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