conditions birational morphism is isomorphism

Question

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?

Answer 1

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...)