Secant varieties of the varieties of reducible hypersurfaces in Pn

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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Singularities and syzygies of secant varieties of nonsingular projective curves

Inventiones mathematicae ◽

10.1007/s00222-020-00976-5 ◽

2020 ◽

Vol 222 (2) ◽

pp. 615-665

Author(s):

Lawrence Ein ◽

Wenbo Niu ◽

Jinhyung Park

Keyword(s):

Secant Varieties ◽

Projective Curves

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Higher secant varieties of the minimal adjoint orbit

Journal of Algebra ◽

10.1016/j.jalgebra.2004.06.009 ◽

2004 ◽

Vol 280 (2) ◽

pp. 743-761 ◽

Cited By ~ 4

Author(s):

Karin Baur ◽

Jan Draisma

Keyword(s):

Secant Varieties ◽

Adjoint Orbit

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On embedded tangent and secant varieties of projective algebraic varieties

Theta Functions–Bowdoin 1987 - Proceedings of Symposia in Pure Mathematics ◽

10.1090/pspum/049.1/1013163 ◽

1989 ◽

pp. 709-718

Author(s):

A. B. Jijtchenko

Keyword(s):

Algebraic Varieties ◽

Secant Varieties

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Secant varieties of toric varieties arising from simplicial complexes

Linear Algebra and its Applications ◽

10.1016/j.laa.2019.12.008 ◽

2020 ◽

Vol 588 ◽

pp. 428-457

Author(s):

M. Azeem Khadam ◽

Mateusz Michałek ◽

Piotr Zwiernik

Keyword(s):

Toric Varieties ◽

Simplicial Complexes ◽

Secant Varieties

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Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties

Canadian Journal of Mathematics ◽

10.4153/cjm-2007-021-6 ◽

2007 ◽

Vol 59 (3) ◽

pp. 488-502 ◽

Cited By ~ 10

Author(s):

A. Bernardi ◽

M. V. Catalisano ◽

A. Gimigliano ◽

M. Idà

Keyword(s):

Hilbert Functions ◽

Secant Varieties ◽

Veronese Varieties ◽

Inverse Systems

AbstractWe consider the k-osculating varieties Ok,n.d to the (Veronese) d-uple embeddings of ℙn. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes Y ⊂ ℙn to and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

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