Let $X \subset \mathbb{P}^n$ be a projective variety and $p \in \mathbb{P}^n$ an arbitrary point not contaned in $X$. The rational projection with center $p$ map $\pi: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ can be described precisely by $x=(x_0:x_1:...:x_n) \mapsto (Y_1(x): ... : Y_n(x))$ where $Y_i = \sum_{j=0}^n a_{ij} X_j$ are linear functions with $p= V(Y_1,...,Y_n)$.
Since one assumes $p \not \in X$ the restriction of $\pi$ to $X$ gives a regular map $p: X \to \mathbb{P}^{n-1}$ with image $I = p(X)$.
Question: Do such restricted projection with center $p$ maps share important general properties as maps maps of varieties? For example is the image $p(X)$ always closed? Or say is $p$ always a closed map? What about smoothness?
Now let consider a special case: Assume that $\operatorname{dim}(X)=n-1$. Let $x \in X$ a smooth point and $p$ is choosen in a way that:
a) $p \not \in X$ as before
b) $p \not \in T_x X \subset \mathbb{P}^n$
The tangent space $T_x X$ in $x$ lives naturally in $\mathbb{P}^n$. Since we assumed $x \in X$ to be smooth, the subvariety $T_x X$ is a projective linear subspace of codimension $1$ (ie isomorphic to $\mathbb{P}^{n-1}$) and now we project away from $p$ onto $T_x X \cong \mathbb{P}^{n-1}$. Restricting the rational projection $\pi: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ as before we obtain a map $p: X \to \mathbb{P}^{n-1}$ which has now some interesting properties. Every fiber equals a line $l \subset \mathbb{P}^n$ through $p$ intersected with $X$. Since $p \not \in X$, the intersection $X \cap l$ is finite and therefore this map is finite.
The important question is why assumptions a) and b) on $p$ also imply that $p: X \to \mathbb{P}^{n-1}$ is etale at the point $x$.
(Source: That's what is clamed in 'Algebraic Curves, Algebraic Manifolds and Schemes' by V.I. Danilov and V.V. Shokurov on page 236). link: https://www.springer.com/gp/book/9783540519959