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Symmetric tensor rank with a tangent vector: a generic uniqueness theorem
Symmetric tensor rank with a tangent vector: a generic uniqueness theorem
- Source :
- Proceedings of the American Mathematical Society, Proceedings of the American Mathematical Society, 2012, 140, 10, pp.3377-3384. ⟨10.1090/S0002-9939-2012-11191-8⟩, Proceedings of the American Mathematical Society, American Mathematical Society, 2012, 140, pp.3377-3384. ⟨10.1090/S0002-9939-2012-11191-8⟩, Ballico, Edoardo ; Bernardi, Alessandra (2011) Symmetric tensor rank with a tangent vector : a generic uniqueness theorem. [Preprint]
- Publication Year :
- 2012
- Publisher :
- HAL CCSD, 2012.
-
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 joint 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 $P_1,...,P_{t-2}\in X_{m,d}$ and a unique tangent vector $\nu$ of $X_{m,d}$ such that $P$ is in the linear span of $\nu \cup \{P_1,...,P_{t-2}\}$, i.e. a degree $d$ linear form $f$ associated to $P$ may be written as $$f = L_{t-1}^{d-1}L_t + \sum_{i=1}^{t-2} L_i^d$$ with $L_i$, $1 \le i \le t$, uniquely determined (up to a constant) linear forms on $\mathbb {P}^m$.<br />Comment: 7 pages
- Subjects :
- MAT/03 Geometria
General Mathematics
010103 numerical & computational mathematics
Rank (differential topology)
01 natural sciences
Linear span
Combinatorics
Mathematics - Algebraic Geometry
Integer
Linear form
FOS: Mathematics
Tangent vector
0101 mathematics
Algebraic Geometry (math.AG)
Mathematics
Degree (graph theory)
Applied Mathematics
14N05, 14M17
010102 general mathematics
Order (ring theory)
weak defectivity
join
VERONESE
Tangential varietie
Veronese variety
tangential variety
Tangent developable
[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Subjects
Details
- Language :
- English
- ISSN :
- 00029939 and 10886826
- Database :
- OpenAIRE
- Journal :
- Proceedings of the American Mathematical Society, Proceedings of the American Mathematical Society, 2012, 140, 10, pp.3377-3384. ⟨10.1090/S0002-9939-2012-11191-8⟩, Proceedings of the American Mathematical Society, American Mathematical Society, 2012, 140, pp.3377-3384. ⟨10.1090/S0002-9939-2012-11191-8⟩, Ballico, Edoardo ; Bernardi, Alessandra (2011) Symmetric tensor rank with a tangent vector : a generic uniqueness theorem. [Preprint]
- Accession number :
- edsair.doi.dedup.....9be6125bc923db2cae29a5470521e01f
- Full Text :
- https://doi.org/10.1090/S0002-9939-2012-11191-8⟩