Let $S$ be an integral Dedekind scheme.


Let $f:X\longrightarrow \mathbf{P}^1\_{S}$ be a finite flat surjective morphism, where $X$ is an integral normal scheme. 


Let $\eta$ be the generic point of $S$. Note that $f\_\eta:X\_\eta\longrightarrow \mathbf{P}^1\_{K(S)}$ is a finite morphism of curves over $K(S)$.


**Question.** Is $f$ the normalization of $\mathbf{P}^1\_S$ in the function field of $X\_\eta$?



Is this morphism the normalization of P^1 in  this curve
In your case the function field of $X\_{\eta}$ is the same as the function field of $X$. Thus the following general remark answers your question affirmatively.


Assume $Y$ is an integral scheme and $L$ is an algebraic extension of the function field $K(Y)$ of $Y$. Let $\pi\colon X \to Y$ be an integral morphism of schemes such that $X$ is integral and normal and such that $\pi$ induces on function fields the extension $K(Y) \subset L = K(X)$. Then $X$ is the normalization of $Y$ in $L$. In fact this follows essentially from the definition of "normalization" and the fact that integral ring homomorphisms are stable under localization.


