Let $L=\mathbb{P}^l\subset\mathbb{P}^N$ be a fixed linear space, $l\geq0$, and let $M=\mathbb{P}^{N-l-1}$ 
be a linear space skew to $L$, i.e. $L\cap M=\emptyset$ and $\langle L, M\rangle=\mathbb{P}^N$. 
Let $X\subseteq\mathbb{P}^N$ be a closed irreducible variety not contained in $L$ 
and let 
$$
\pi\_L:X\dashrightarrow\mathbb{P}^{N-l-1}=M
$$
be the linear projection, i.e. the rational map defined on $X\setminus(L\cap X)$ by 
$$
\pi\_L(x)=\langle L,x\rangle\cap M.
$$


I say that (denoting by $x$ the general point of $X$):


* $\pi\_L$ is generically quasi-finite if $\pi\_L^{-1}(\pi\_L(x))$ is a finite set;
* $\pi\_L$ is generically unramified if $\pi\_L^{-1}(\pi\_L(x))$ coincide, as a scheme, with the point $x$ in a neighbourhood of $x$.


Is it true that if $\pi\_L$ is generically quasi-finite then it's generically unramified ?



On quasi-finite and unramified linear projections.
The answer is **yes** in characteristic $0$.


In fact, take the non-empty Zariski open set $X^0$ where $\pi \colon X \to M$ has finite fibres, and let $M^0 \subset M$ be the image of $X^0$. Then $M^0$ is a Zariski open set of $M$ and the restriction $\pi^0 \colon X^0 \to M^0$ is a finite map. Passing to function fields, this means that
$$(\pi^{0})^\*(K(M^0)) \subset K(X^0)$$
is a *finite* field extension, and since we are working in characteristic $0$ it is also *separable*.


Now we can apply the following result, see [Shafarevich, Basic Algebraic Geometry I, Theorem 4 of Chapter II, page 144]:



> 
> **Theorem.** The set of points where a finite map $f \colon X \to Y$ is unramified is open, and it is nonempty if $f^\*(K(Y)) \subset K(X)$ is a separable field extension.
> 
> 
> 


It follows that $\pi^0 \colon X^0 \to M^0$ is generically unramified, so a fortiori $\pi \colon X \to M$ is generically unramified.


If the characteristic of the base field is $p >0$ the result is no longer true, as it is shown by MP's comment about strange curves. 


