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 $\tau (X_{m,d})\subset \mathbb {P}^N$, 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 $$f = L_{t-1}^{d-1}L_t + \sum _{i=1}^{t-2} L_i^d, \; \; \; \; (T = v_{t-1}^{\otimes (d-1)}v_t + \sum _{i=1}^{t-2} v_i^{\otimes d})$$ 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).