# ag.algebraic geometry – A question on linear projection of a smooth projective variety

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 $$L subset mathbb{P}^{N_d}$$ of dimension $$N_d-n-2$$, not intersecting $$sigma_d(X)$$ such that for the linear projection from $$L$$ (sometimes called projection with centre $$L$$):
$$pi_L : sigma_d(X) to mathbb{P}^{n+1}$$ we have $$pi_L^{-1}(pi_L(sigma_d(x)))=sigma_d(x)$$ i.e., the preimage of $$sigma_d(x)$$ is only $$sigma_d(x)$$? Any hint/reference is most welcome.