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

Equations for secant varieties of Chow varieties

International Journal of Algebra and Computation ◽

10.1142/s0218196717500515 ◽

2017 ◽

Vol 27 (08) ◽

pp. 1087-1111

Author(s):

Yonghui Guan

Keyword(s):

Secant Varieties ◽

Linear Forms ◽

Product Of Linear Forms ◽

Chow Variety ◽

The Ideal

The Chow variety of polynomials that decompose as a product of linear forms has been studied for more than 100 years. Finding equations in the ideal of secant varieties of Chow varieties would enable one to prove Valiant's conjecture [Formula: see text]. In this paper, I use the method of prolongation to obtain equations for secant varieties of Chow varieties as [Formula: see text]-modules.

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Singular Poisson–Kähler geometry of Scorza varieties and their secant varieties

Differential Geometry and its Applications ◽

10.1016/j.difgeo.2005.03.007 ◽

2005 ◽

Vol 23 (1) ◽

pp. 79-93 ◽

Cited By ~ 1

Author(s):

Johannes Huebschmann

Keyword(s):

Secant Varieties ◽

Kahler Geometry

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Varieties with an extremal number of degenerate higher secant varieties.

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1988.392.16 ◽

1988 ◽

Vol 1988 (392) ◽

pp. 16-26 ◽

Cited By ~ 4

Keyword(s):

Secant Varieties

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Non-defectivity of Grassmannian bundles over a curve

International Journal of Mathematics ◽

10.1142/s0129167x16400024 ◽

2016 ◽

Vol 27 (07) ◽

pp. 1640002 ◽

Cited By ~ 1

Author(s):

Insong Choe ◽

George H. Hitching

Keyword(s):

Vector Bundles ◽

Manuscripta Math ◽

Maximal Degree ◽

Geometric Proof ◽

Secant Varieties ◽

Grassmann Bundle ◽

Positive Rank

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

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The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics ◽

10.3390/math6120314 ◽

2018 ◽

Vol 6 (12) ◽

pp. 314 ◽

Cited By ~ 8

Author(s):

Alessandra Bernardi ◽

Enrico Carlini ◽

Maria Catalisano ◽

Alessandro Gimigliano ◽

Alessandro Oneto

Keyword(s):

Algebraic Geometry ◽

Projective Variety ◽

Tensor Decomposition ◽

Tensor Rank ◽

Segre Variety ◽

Secant Varieties ◽

Open Problems ◽

Veronese Variety ◽

Strong Motivation

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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