Let $X$ and $Y$ be two varieties, and $f:X\rightarrow Y$ be a morphism. Suppose moreover there is a point $Q\in Y$ and $P=f^{-1}(Q)\in X$, such that the restriction $f:X\setminus P\rightarrow Y\setminus Q$ is an isomorphism. So in particular $f$ is a bijective birational morphism.
I was wondering under which conditions $f$ is an isomorphism. It is certainly not true in general. For example if $X$ is the affine line, $Y$ the cusp and $f:X\rightarrow Y: t\mapsto (t^3, t^2)$. Then $f: X\setminus 0\rightarrow Y\setminus (0,0)$ is an isomorphism, but of course $f:X\rightarrow Y$ is not an isomorphism.
Is it true if $X$ and $Y$ are smooth? Or if $X$ and $Y$ are projective?
Best Answer
Projective will not be enough to make it work, because your example of the cusp can also arise for projective curves.
On the other hand, as long as $Y$ is normal and $f$ is proper, then it is true, by a version of Zariski's Main Theorem (Liu, 4.4.3).
Edit: As QiL'8 explains, the properness hypothesis is unnecessary. (Perhaps in future I should leave these things to the expert...)