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

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 $τ(X_{m,d})\subset \mathbb {P}^N$, be the tangent developable of $X_{m,d}$. For each integer $t \ge 2$ let $τ(X_{m,d},t)\subseteq \mathbb {P}^N$, be the joint of $τ(X_{m,d})$ 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 $P\in τ(X_{m,d},t)$ there are uniquely determined $P_1,...,P_{t-2}\in X_{m,d}$ and a unique tangent vector $ν$ of $X_{m,d}$ such that $P$ is in the linear span of $ν\cup \{P_1,...,P_{t-2}\}$, 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 $\mathbb {P}^m$.

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