TITLE: Does generically finite imply quasi-finite?
QUESTION [1 upvotes]: If $f: X \rightarrow Y$ is a morphism of $k$-varieties ($k$ a field, characteristic 0 or even algebraically closed if it matters), and $f$ is dominant and generically finite, then is $f$ quasi-finite? By quasi-finite I mean that for each $y \in Y$ $f^{-1}(y)$ is finite.

REPLY [2 votes]: No. Consider $Bl_{(0,0)} \Bbb A^2\to \Bbb A^2$, the blowup of the origin in $\Bbb A^2$. This is an isomorphism on $\Bbb A^2\setminus \{(0,0)\}$ but has a whole $\Bbb P^1$ over the origin.
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