On singularities of third secant varieties of Veronese embeddings

Han, Kangjin

doi:10.1016/j.laa.2018.01.013

Cited by 5 publications

(7 citation statements)

References 11 publications

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“…This extends the results for the Segre [27] and the Veronese [18] cases. (For secant varieties of the Veronese the singular locus was described in small cases in [16].) We also provide, in Theorem 4.4 and Theorem 4.6, a complete classification of the secant varieties of Segre-Veronese varieties that are (Q−) Gorenstein, extending previous results for matrices and Segre products.…”

Section: Introductionsupporting

confidence: 61%

Secant varieties of toric varieties arising from simplicial complexes

Khadam

Michałek

Zwiernik

2020

Linear Algebra and its Applications

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Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such secant is toric, which gives a way to use combinatorial tools to study singularities. We focus on the Segre-Veronese variety for which we completely classify their secants that give Gorenstein or Q-Gorenstein varieties. We conclude providing the explicit description of the singular locus.

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Section: Introductionsupporting

confidence: 61%

Secant varieties of toric varieties arising from simplicial complexes

Khadam

Michałek

Zwiernik

2020

Linear Algebra and its Applications

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“…(iv) For (n, k) = (2, 3), smoothness of all points in σ 3 (ν d (P 1 )) \ σ 2 (ν d (P 3 )) for d ≥ 4 was already proved in [Han18,theorem 2.14]. This is included for completeness of the statement.…”

Section: Introductionmentioning

confidence: 80%

“…(a) In the case of k = 3 and n ≥ 3, the condition of Part (ii) in Theorem 1 holds if and only if d = 4, and then it holds Sing(σ 3 (v 4 (P n )) = P 1 ⊂P n v 4 (P 1 ) since σ 3 (v 4 (P 1 )) = v d (P 1 ) and σ 2 (v 4 (P n )) ⊂ P 1 ⊂P n v d (P 1 ) (see also Corollary 24). This gives a geometric description of the only exceptional case on the singular loci of 3-th secant variety in [Han18]. As an application of our main results, we also consider the case of singular loci of the fourth-secant variety of any Veronese embeddings.…”

Section: Introductionmentioning

confidence: 88%

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On the singular loci of higher secant varieties of Veronese embeddings

Furukawa¹,

Han²

2021

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In this paper we study singular loci of higher secant varieties of the image of the d-uple Veronese embedding of projective n-space, ν d (P n ) ⊂ P N=( n+d d )−1 . We call the closure of the union of all the (k − 1)-planes spanned by k points on X the k-th secant of X for a given embedded variety X ⊂ P N and denote it by σ k (X). For the singular loci of σ k (ν d (P n )), it has been known only for k ≤ 3. In the present paper, as investigating geometry of moving tangents along subvarieties, we determine the (non-)singularity of so-called subsecant loci of σ k (ν d (P n )) for arbitrary k. We also consider the case of the 4-th secant of ν d (P n ) by applying main results and computing conormal space via a certain type of Young flattening. In the end, some discussion for further development is presented as well.

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“…By Proposition 5.2, the required fundamental group is Z/2Z. (iii) When r = 3, the singular locus of σ 3 (X) is σ 2 (X) by [20]. As d > 3, for any x ∈ σ 2 (X), we must have rank S (x) = 3, which implies that σ 2 (X) ∩ s 3 (X) = s 2 (X).…”

Section: Higher-order Connectedness Of Symmetric Tensor Rankmentioning

confidence: 99%

Topology of tensor ranks

Comon

Lim

Yang

et al. 2020

Advances in Mathematics

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We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over C, the set of rank-r tensors and the set of symmetric rank-r symmetric tensors are both path-connected if r is not more than the complex generic rank; these results also extend to border rank and symmetric border rank over C. Over R, the set of rank-r tensors is path-connected if it has the expected dimension but the corresponding result for symmetric rank-r symmetric dtensors depends on the order d: connected when d is odd but not when d is even. Border rank and symmetric border rank over R have essentially the same path-connectedness properties as rank and symmetric rank over R. When r is greater than the complex generic rank, we are unable to discern any general pattern: For example, we show that border-rank-three tensors in R 2 ⊗ R 2 ⊗ R 2 fall into four connected components. For multilinear rank, the manifold of d-tensors of multilinear rank (r1, . . . , r d ) in C n 1 ⊗· · ·⊗C n d is always path-connected, and the same is true in R n 1 ⊗· · ·⊗R n d unless ni = ri = j =i rj for some i ∈ {1, . . . , d}. Beyond path-connectedness, we determine, over both R and C, the fundamental and higher homotopy groups of the set of tensors of a fixed small rank, and, taking advantage of Bott periodicity, those of the manifold of tensors of a fixed multilinear rank. We also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank.2010 Mathematics Subject Classification. 15A69, 54D05, 55Q05.

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On singularities of third secant varieties of Veronese embeddings

Abstract: In this paper we study singularities of third secant varieties of Veronese embedding v d (P n ), which corresponds to the variety of symmetric tensors of border rank at most three in (C n+1 ) ⊗d .

Cited by 5 publications

References 11 publications

Secant varieties of toric varieties arising from simplicial complexes

Secant varieties of toric varieties arising from simplicial complexes

On the singular loci of higher secant varieties of Veronese embeddings

Topology of tensor ranks

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