scholarly journals The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 314 ◽  
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.

scholarly journals Tangent loci and certain linear sections of adjoint varieties

2000 ◽  
Vol 158 ◽  
pp. 63-72
Author(s):  
Hajime Kaji ◽  
Osami Yasukura
Keyword(s):  
Lie Algebra ◽  
Projective Variety ◽  
Linear Subspace ◽  
Nilpotent Orbit ◽  
Contact Type ◽  
Simple Lie Algebra ◽  
Veronese Variety ◽  
Linear Sections

AbstractAn adjoint variety X(g)associated to a complex simple Lie algebra is by definition a projective variety in ℙ*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X(g) in terms of triples. Secondly for a graded decomposition of contact type we show that the intersection of X(g) and the linear subspace ℙ*(g1) in ℙ*(g) coincides with the cubic Veronese variety associated to g.

Tensor Space

2011 ◽  
pp. 135-149
Author(s):  
David Zhang ◽  
Fengxi Song ◽  
Yong Xu ◽  
Zhizhen Liang
Keyword(s):  
Tensor Decomposition ◽  
Decomposition Methods ◽  
Tensor Rank ◽  
Tensor Space

In this chapter, we first give the background materials for developing tensor discrimination technologies in Section 7.1. Section 7.2 introduces some basic notations in tensor space. Section 7.3 discusses several tensor decomposition methods. Section 7.4 introduces the tensor rank.

scholarly journals Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

2015 ◽  
Vol 2015 (702) ◽  
pp. 1-40 ◽  
Author(s):  
Timo Schürg ◽  
Bertrand Toën ◽  
Gabriele Vezzosi
Keyword(s):  
Algebraic Geometry ◽  
Projective Variety ◽  
Obstruction Theory ◽  
Important Ingredient ◽  
Smooth Complex ◽  
Perfect Complexes ◽  
Derived Algebraic Geometry ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractA quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety.ForWe give two further applications toAn important ingredient of our construction is a

scholarly journals Geometry of Higher-Order Markov Chains

2012 ◽  
Vol 3 (1) ◽  
Author(s):  
Bernd Sturmfels
Keyword(s):  
Markov Chains ◽  
Prime Ideal ◽  
Linear Space ◽  
Projective Variety ◽  
Likelihood Function ◽  
Distribution Model ◽  
Segre Variety ◽  
Linear Forms ◽  
Transition Distribution ◽  
Binary Case

We determine an explicit Grobner basis, consisting of linear forms and determinantalquadrics, for the prime ideal of Raftery's mixture transition distribution model for Markov chains.When the states are binary, the corresponding projective variety is a linear space, the model itselfconsists of two simplices in a cross-polytope, and the likelihood function typically has two localmaxima. In the general non-binary case, the model corresponds to a cone over a Segre variety.

scholarly journals ON DIFFERENCES BETWEEN THE BORDER RANK AND THE SMOOTHABLE RANK OF A POLYNOMIAL

2014 ◽  
Vol 57 (2) ◽  
pp. 401-413 ◽  
Author(s):  
WERONIKA BUCZYŃSKA ◽  
JAROSŁAW BUCZYŃSKI
Keyword(s):  
General Point ◽  
Secant Varieties ◽  
Secant Variety ◽  
Veronese Variety ◽  
Border Rank ◽  
Veronese Varieties ◽  
Special Points

AbstractWe consider higher secant varieties to Veronese varieties. Most points on the rth secant variety are represented by a finite scheme of length r contained in the Veronese variety – in fact, for a general point, the scheme is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note, we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.

scholarly journals On the Terracini Locus of Projective Varieties

2021 ◽  
Author(s):  
Edoardo Ballico ◽  
Luca Chiantini
Keyword(s):  
Linear System ◽  
Symmetric Product ◽  
Secant Varieties ◽  
Finite Sets ◽  
Projective Varieties ◽  
Special Position ◽  
Double Points ◽  
Veronese Variety ◽  
Interpolation Problems

AbstractWe introduce and study properties of the Terracini locus of projective varieties X, which is the locus of finite sets $$S \subset X$$ S ⊂ X such that 2S fails to impose independent conditions to a linear system L. Terracini loci are relevant in the study of interpolation problems over double points in special position, but they also enter naturally in the study of special loci contained in secant varieties to projective varieties.We find some criteria which exclude that a set S belongs to the Terracini locus. Furthermore, in the case where X is a Veronese variety, we bound the dimension of the Terracini locus and we determine examples in which the locus has codimension 1 in the symmetric product of X.

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