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Let $f: Y \to X$ be a birational proper morphism. Assume that every point of $X$ has an etale neighbourhood over which $f$ has a section. Is it true that $f$ is an isomorphism?

Adam
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  • No, what about $Y=\mathbb{P}^1\sqcup\mathbb{P}^1$ and $X=\mathbb{P}^1$? Am I misunderstanding your question? – Alex Youcis Aug 21 '14 at 21:40
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    The question is local on $X$, and by faithfully flat descent, one can suppose that $f$ has a section. This implies that $f$ is an isomorphism. @AlexYoucis: $f$ is not birational in your example. – Cantlog Aug 21 '14 at 22:15
  • @Cantlog Oh, of course. How silly of me. Thanks :) – Alex Youcis Aug 22 '14 at 10:51
  • @Cantlog: I suggest you turn your comment into an answer. Just to be sure, by "local" you mean Zariski, right? It may also be helpful to Adam to flesh out your use of faithfully flat descent (or provide a reference). – RghtHndSd Aug 22 '14 at 20:01
  • @RghtHndSd: yes I should be more precise, thanks "The question is Zariski local on $X$". But we could probably avoid fppf descent because the existence of an etale neighborhood with a section implies immediately (set-theoretical) that $f$ is quasi-finite, hence finite and birational. Moreover, $Y\to X$ is dominated by an etale morphism. Maybe this is enough to imply that $Y\to X$ is an isomorphism. – Cantlog Aug 22 '14 at 22:32

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