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

Part of: Cycles and subschemes Basic linear algebra Computational aspects in algebraic geometry Special varieties

Published online by Cambridge University Press:  17 December 2014

WERONIKA BUCZYŃSKA  and
JAROSŁAW BUCZYŃSKI
WERONIKA BUCZYŃSKA
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland e-mails: wkrych@mimuw.edu.pl; jabu@mimuw.edu.pl
JAROSŁAW BUCZYŃSKI
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland e-mails: wkrych@mimuw.edu.pl; jabu@mimuw.edu.pl
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Abstract

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We 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.

MSC classification

Primary: 14M17: Homogeneous spaces and generalizations
Secondary: 14C05: Parametrization (Chow and Hilbert schemes) 15A69: Multilinear algebra, tensor products 14Q20: Effectivity, complexity
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 2 , May 2015 , pp. 401 - 413
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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