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 ?

strange curvesof Hartshorne's Definition on page 311 would give you problems. In particular, if $C$ is a smooth plane conic in characteristic 2 there is a point $p$ in the plane contained in all tangent lines to $C$. In this case, choosing $L=p$, all fibers are generically quasi-finite, but they are all non-reduced, so that the morphism is not generically unramified. It is true though that smooth conics in characteristic 2 are (essentially) the only strange curves... – M P Sep 6 '11 at 8:12