Symmetric tensor rank with a tangent vector: a generic uniqueness theorem
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- by Edoardo Ballico and Alessandra Bernardi PDF
- Proc. Amer. Math. Soc. 140 (2012), 3377-3384 Request permission
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 \[ f = L_{t-1}^{d-1}L_t + \sum _{i=1}^{t-2} L_i^d,\quad (T = v_{t-1}^{\bigotimes (d-1)}v_t + \sum _{i=1}^{t-2} v_i^{\bigotimes d})\] 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
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Additional Information
- Edoardo Ballico
- Affiliation: Department of Mathematics, University of Trento, 38123 Povo (TN), Italy
- MR Author ID: 30125
- Email: ballico@science.unitn.it
- Alessandra Bernardi
- Affiliation: GALAAD, INRIA Méditerranée, BP 93, 06902 Sophia Antipolis, France
- Email: alessandra.bernardi@inria.fr
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3377-3384
- MSC (2010): Primary 14N05, 14M17
- DOI: https://doi.org/10.1090/S0002-9939-2012-11191-8
- MathSciNet review: 2929007