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Let $X$ and $Y$ be normal projective varieties. Let $\pi:X\to Y$ be a finite surjective morphism and $\tau:X\to Y$ a birational morphism.

Question: will $\tau$ be isomorphic? or any counter-example?

(Note that if $\deg \pi=1$, then $\tau$ is isomorphic.)

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    $\begingroup$ The blow down of a $(-1)$-curve is a counter example. Maybe there is an error in the question, because the statement in brackets doesn't make sense i.e. there is no relation between $\pi$ and $\tau$. $\endgroup$
    – Nick L
    Commented Jun 5, 2019 at 10:59
  • $\begingroup$ @NickL Thanks, I see! $\endgroup$
    – Sheng Meng
    Commented Jun 5, 2019 at 11:03
  • $\begingroup$ @NickL If $\pi$ is isomorphic, then $\tau:X\to Y\cong X$ is a birational automorphism which has to be an automorphism. $\endgroup$
    – Sheng Meng
    Commented Jun 5, 2019 at 11:15
  • $\begingroup$ @NickL The original question has many other assumptions, I just wonder what happens if removing all the assumptions. I would appreciate you copy this comment to an answer, I guess you mean $X:=$ blowup of $\mathbb{P}^2$, right? It has finite morphism to $\mathbb{P}^2$. $\endgroup$
    – Sheng Meng
    Commented Jun 5, 2019 at 11:25
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    $\begingroup$ Take $Y=\mathbb{P}^2$, $X=$ a del Pezzo surface of degree 4. There is a double covering $\pi : X\rightarrow \mathbb{P}^2$ and a birational morphism $\tau :X\rightarrow \Bbb{P}^2$ obtained by blowing up 7 points in general position. $\endgroup$
    – abx
    Commented Jun 5, 2019 at 14:13

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