Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $n$. Fix a closed point $x \in X$ and an embedding of $X$ in $\mathbb{P}^m$ for some integer $m$. For a given $d$, denote by $\sigma_d : \mathbb{P}^m \to \mathbb{P}^{N_d}$ the $d$-tuple embedding. My question is: for $d \gg 0$, does there exist a linear subspace $L_x \subset \mathbb{P}^{N_d}$ (depending on $x$) of dimension $N_d-n-2$, not intersecting $\sigma_d(X)$ such that for the linear projection from $L_x$ (sometimes called *projection with centre* $L$):
$$\pi_{L_x} : \sigma_d(X) \to \mathbb{P}^{n+1}$$ we have $\pi_{L_x}^{-1}(\pi_{L_x}(\sigma_d(x)))=\sigma_d(x)$ i.e., the preimage of $\sigma_d(x)$ is only $\sigma_d(x)$ **for the chosen closed point $x$**? Any hint/reference is most welcome.

**EDIT** Note that, the choice of $L_x$ depends on the choice of $x$.