I’m trying to understand the proof that all quasi-projective varieties have a normalisation, starting from the result that all affine varieties have a normalisation, but I’m skipping a bunch of details which I’m not sure they will be relevant.

To the point: Let $U$ be an affine curve and let $hat U$ be its normalisation, i.e., we have a finite and birational morphism $n: hat U rightarrow U$. The proof takes an open set $V subseteq U$ and claims that $n: n ^{-1} (V) rightarrow V$ is still birational (which is clear) and finite, so it is its normalisation. My question is, this does not happen necessarily for ANY open $V$, right? It must’ve been a special $V$, so, does it have to be affine? does it have be such that $V$ and $n ^{-1} V$ are both affine (can I always reduce my open set to find one like this)? If either of these hypotheses, or in fact no hypothesis at all, is enough, could you help me in showing it?

**Tiny summary**: if $n : hat U rightarrow U$ is a finite and birational morphism (with both $hat U$ and $U$ algebraic curves), and $V subseteq U$ some open set, what do I have to assume about $V$ so that $n : n ^{-1} (V) rightarrow V$ is still finite?