Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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).

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14N05, 14M17

Retrieve articles in all journals with MSC (2010): 14N05, 14M17


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/S0002-9939-2012-11191-8
PII: S 0002-9939(2012)11191-8
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.