scholarly journals Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties

2007 ◽  
Vol 59 (3) ◽  
pp. 488-502 ◽  
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.

scholarly journals Secant varieties of Segre–Veronese varieties

2012 ◽  
Vol 6 (8) ◽  
pp. 1817-1868 ◽  
Author(s):  
Claudiu Raicu
Keyword(s):  
Secant Varieties ◽  
Veronese Varieties

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.

About the Defectivity of Certain Segre–Veronese Varieties

2008 ◽  
Vol 60 (5) ◽  
pp. 961-974 ◽  
Author(s):  
Silvia Abrescia
Keyword(s):  
Secant Varieties ◽  
Secant Variety ◽  
Determinantal Equation ◽  
Veronese Varieties

AbstractWe study the regularity of the higher secant varieties of ℙ1 × ℙn, embedded with divisors of type (d, 2) and (d, 3). We produce, for the highest defective cases, a “determinantal” equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of ℙn is not Grassmann defective.

scholarly journals Moment Varieties of Gaussian Mixtures

2016 ◽  
Vol 7 (1) ◽  
Author(s):  
Carlos Amendola ◽  
Jean-Charles Faugere ◽  
Bernd Sturmfels
Keyword(s):  
Maximum Likelihood ◽  
Probability Distributions ◽  
Gaussian Mixtures ◽  
Secant Varieties ◽  
Maximum Likelihood Approach ◽  
Veronese Varieties ◽  
Classical Work ◽  
Likelihood Approach ◽  
The Moment

The points of a moment variety are the vectors of all moments up to some order, for a givenfamily of probability distributions. We study the moment varieties for mixtures of multivariate Gaussians.Following up on Pearson's classical work from 1894, we apply current tools from computational algebrato recover the parameters from the moments. Our moment varieties extend objects familiar to algebraicgeometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting allcovariance matrices to zero. We compute the ideals of the 5-dimensional moment varieties representingmixtures of two univariate Gaussians, and we oer a comparison to the maximum likelihood approach.

scholarly journals New examples of defective secant varieties of Segre–Veronese varieties

2011 ◽  
Vol 63 (3) ◽  
pp. 287-297 ◽  
Author(s):  
Hirotachi Abo ◽  
Maria Chiara Brambilla
Keyword(s):  
Secant Varieties ◽  
Veronese Varieties

Secant Varieties of Segre–Veronese Varieties ℙm× ℙnEmbedded byO(1, 2)

2009 ◽  
Vol 18 (3) ◽  
pp. 369-384 ◽  
Author(s):  
Hirotachi Abo ◽  
Maria Chiara Brambilla
Keyword(s):  
Secant Varieties ◽  
Veronese Varieties

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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