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Let $X$ be an integral separated scheme of finite type over $\mathbb{C}$. Consider the normalization morphism $f:X'\rightarrow X$. Can we always find an affine open $U\subset X'$ such that $f|_U:U\rightarrow X$ is bijective on the underlying topological spaces?

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  • $\begingroup$ Consider $\mathbb{P}^1$. Though I do not know if what you say ever happens for $X$ non-affine. That might be an interesting question (or it might be me being stupid). $\endgroup$ – user138661 May 2 at 11:25
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    $\begingroup$ What if $f$ is already bijective? (e.g., $X$ a curve with a cusp). $\endgroup$ – abx May 2 at 11:36
  • $\begingroup$ @abx but do you know an example with $X$ non-affine, receiving a map from an affine scheme as described in the question? I am genuinely curious. $\endgroup$ – user138661 May 2 at 11:57
  • $\begingroup$ @schematic_boi: Take $X$ a cubic in $\mathbb{P}^2$ with an ordinary double point $s$, with normalization $f: \mathbb{P}^1\rightarrow X$ such that $f(0)=s$, and $U=\mathbb{P}^1\smallsetminus\{0\} $. $\endgroup$ – abx May 2 at 14:35

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