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?

Note that in positive characteristic, bijective doesn't imply birational (e.g. the Frobenius morphism). — , Apr 13 '13 at 20:39

score 6 · Accepted Answer · 2013-04-13T20:36:32.130

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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...)

edited Apr 13 '13 at 20:36

answered Apr 13 '13 at 17:46

in fact, this is true without assuming f proper. – Apr 13 '13 at 20:17
@QiL'8: Yes, I thought so, but couldn't quite put the pieces together in that case. Can you explain? – Apr 13 '13 at 20:22
because by ZMT, a quasi-finite morphism to a normal noetherian scheme is an open immersion. – Apr 13 '13 at 20:30
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(Quasi-finite birational?) Of course! – Apr 13 '13 at 20:33
Absolutely, thanks. – Apr 13 '13 at 20:35
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I disagree with the second part of edit. – Apr 13 '13 at 20:38

conditions birational morphism is isomorphism

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