Ballico, Edoardo ; Bernardi, Alessandra
(2011)
*Symmetric tensor rank with a tangent vector : a generic uniqueness theorem.*
[Preprint]

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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 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 $P_1,\dots ,P_{t-2}\in X_{m,d}$ 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 $f$ 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 $\mathbb {P}^m$.

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 $P_1,\dots ,P_{t-2}\in X_{m,d}$ 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 $f$ 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 $\mathbb {P}^m$.

Document type

Preprint

Creators

Keywords

Veronese variety; tangential variety; join; weak defectivity

Subjects

DOI

Deposit date

01 Mar 2011 11:04

Last modified

16 May 2011 12:17

URI

## Other metadata

Document type

Preprint

Creators

Keywords

Veronese variety; tangential variety; join; weak defectivity

Subjects

DOI

Deposit date

01 Mar 2011 11:04

Last modified

16 May 2011 12:17

URI

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