# Bijective restriction of the normalization morphism

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?

• 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). – user138661 May 2 at 11:25
• What if $f$ is already bijective? (e.g., $X$ a curve with a cusp). – abx May 2 at 11:36
• @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. – user138661 May 2 at 11:57
• @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 $U=\mathbb{P}^1\smallsetminus\{0\}$. – abx May 2 at 14:35