Let $E$ be a curve (so a $1$-dimensional, proper k-scheme) which is irreducible.
Let be $f:E \to \mathbb{P}^1$ be a birational morphism.
Why it is then an isomorphism?
Does it something to do with the fact that $\mathbb{P}^1$ is normal?
My thoughts:
Since $E$ is proper the morphism is closed. Futhermore, by birationality, is must be surjective. From here I surrender. Especially how to show this isomorphism on level of stalks.
I guess that using nomality of $\mathbb{P}^1$ (therefore normality of stalks as local rings) it suffies to show that stalk maps $\mathcal{O}_{{\mathbb{P}^1},f(a)} \to \mathcal{O}_{E,a}$ are integral, but how.