Higher distances for constant dimensions codes: the case of osculating spaces to a Veronese variety

AbstractWe compute the equation of the 7-secant variety to the Veronese variety (P4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariant of plane cubics as a pfaffian.

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Tangent loci and certain linear sections of adjoint varieties

Nagoya Mathematical Journal ◽

10.1017/s0027763000007297 ◽

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.

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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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Osculating Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-057-9 ◽

1962 ◽

Vol 14 ◽

pp. 669-684

Author(s):

Peter Scherk

Keyword(s):

Duality Theorem ◽

Characteristic Curve ◽

Independent Interest ◽

Natural Approach ◽

Osculating Spaces ◽

Analytical Tools

In this paper an attempt is made to prove some of the basic theorems on the osculating spaces of a curve under minimum assumptions. The natural approach seems to be the projective one. A duality yields the corresponding results for the characteristic spaces of a family of hyperplanes. A duality theorem for such a family and its characteristic curve also is proved. Finally the results are applied to osculating hyperspheres of curves in a conformai space.The analytical tools are collected in the first three sections. Some of them may be of independent interest.

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