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Symmetric tensor rank with a tangent vector: a generic uniqueness theorem

Authors: Edoardo Ballico and Alessandra Bernardi
Journal: Proc. Amer. Math. Soc. 140 (2012), 3377-3384
MSC (2010): Primary 14N05, 14M17
Published electronically: February 22, 2012
MathSciNet review: 2929007
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Abstract: Let $ X_{m,d}\subset \mathbb{P}^N$, $ N:= \binom {m+d}{m}-1$, be the order $ d$ Veronese embedding of $ \mathbb{P}^m$. Let $ \tau (X_{m,d})\subset \mathbb{P}^N$ be the tangent developable of $ X_{m,d}$. For each integer $ t \ge 2$ let $ \tau (X_{m,d},t)\subseteq \mathbb{P}^N$ be the join of $ \tau (X_{m,d})$ and $ t-2$ copies of $ X_{m,d}$. Here we prove that if $ m\ge 2$, $ d\ge 7$ and $ t \le 1 + \lfloor \binom {m+d-2}{m}/(m+1)\rfloor $, then for a general $ P\in \tau (X_{m,d},t)$ there are uniquely determined $ P_1,\dots ,P_{t-2}\in X_{m,d}$ and a unique tangent vector $ \nu $ of $ X_{m,d}$ such that $ P$ is in the linear span of $ \nu \cup \{P_1,\dots ,P_{t-2}\}$; i.e. a degree $ d$ linear form $ f$ (a symmetric tensor $ T$ of order $ d$) associated to $ P$ may be written as

$\displaystyle f = L_{t-1}^{d-1}L_t + \sum _{i=1}^{t-2} L_i^d,\quad (T = v_{t-1}^{\bigotimes (d-1)}v_t + \sum _{i=1}^{t-2} v_i^{\bigotimes d})$

with $ L_i$ linear forms on $ \mathbb{P}^m$ ($ v_i$ vectors over a vector field of dimension $ m+1$ respectively), $ 1 \le i \le t$, that are uniquely determined (up to a constant).

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Additional Information

Edoardo Ballico
Affiliation: Department of Mathematics, University of Trento, 38123 Povo (TN), Italy

Alessandra Bernardi
Affiliation: GALAAD, INRIA Méditerranée, BP 93, 06902 Sophia Antipolis, France

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 Mittag-Leffler (Sweden), Marie Curie: Promoting Science (FP7-PEOPLE-2009-IEF), 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.