Symmetric tensor rank with a tangent vector: a generic uniqueness theorem
Authors:
Edoardo Ballico and Alessandra Bernardi
Journal:
Proc. Amer. Math. Soc. 140 (2012), 33773384
MSC (2010):
Primary 14N05, 14M17
Published electronically:
February 22, 2012
MathSciNet review:
2929007
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Abstract: Let , , be the order Veronese embedding of . Let be the tangent developable of . For each integer let be the join of and copies of . Here we prove that if , and , then for a general there are uniquely determined and a unique tangent vector of such that is in the linear span of ; i.e. a degree linear form (a symmetric tensor of order ) associated to may be written as with linear forms on ( vectors over a vector field of dimension respectively), , that are uniquely determined (up to a constant).
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 B. Ådlandsvik, Joins and higher secant varieties. Math. Scand. 61 (1987), no. 2, 213222. MR 947474 (89j:14030)
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 E. Ballico, On the weak nondefectivity of Veronese embeddings of projective spaces. Cent. Eur. J. Math. 3 (2005), no. 2, 183187. MR 2129920 (2005m:14097)
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 E. Ballico, A. Bernardi, Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes, arXiv:1010.3546v1 [math.AG].
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 A. Bernardi, A. Gimigliano, M. Idà. Computing symmetric rank for symmetric tensors.
J. Symb. Comput. 46 (2011) 3453. MR 2736357
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 M. V. Catalisano, A. V. Geramita, A. Gimigliano, On the secant varieties to the tangential varieties of a Veronesean. Proc. Amer. Math. Soc. 130 (2002), no. 4, 975985. MR 1873770 (2002m:14042)
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 C. Ciliberto, F. Russo, Varieties with minimal secant degree and linear systems of maximal dimension on surfaces. Adv. Math. 206 (2006), no. 1, 150. MR 2199628 (2007d:14097)
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 M. Mella, Singularities of linear systems and the Waring problem. Trans. Amer. Math. Soc. 358 (2006), no. 12, 55235538. MR 2238925 (2007h:14059)
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 M. Mella, Base loci of linear systems and the Waring problem. Proc. Amer. Math. Soc. 137 (2009), no. 1, 9198. MR 2439429 (2009g:14072)
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Additional Information
Edoardo Ballico
Affiliation:
Department of Mathematics, University of Trento, 38123 Povo (TN), Italy
Email:
ballico@science.unitn.it
Alessandra Bernardi
Affiliation:
GALAAD, INRIA Méditerranée, BP 93, 06902 Sophia Antipolis, France
Email:
alessandra.bernardi@inria.fr
DOI:
http://dx.doi.org/10.1090/S000299392012111918
Keywords:
Veronese variety,
tangential variety,
join,
weak defectivity
Received by editor(s):
January 26, 2011
Received by editor(s) in revised form:
April 11, 2011
Published electronically:
February 22, 2012
Additional Notes:
The authors were partially supported by CIRM of FBK Trento (Italy), Project Galaad of INRIA Sophia Antipolis Méditerranée (France), Institut MittagLeffler (Sweden), Marie Curie: Promoting Science (FP7PEOPLE2009IEF), MIUR and GNSAGA of INdAM (Italy).
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
