Let $f:X\longrightarrow S$ be a flat projective morphism of regular integral noetherian schemes such that that the generic fibre $X\_\eta\longrightarrow K(S)$ is a smooth projective connected curve over $K(S)$.


*Assumptions.* For simplicity, let us assume that $S$ is one-dimensional, i.e., $S$ is a connected Dedekind scheme and that $K(S)$ is perfect. (I don't think we need these assumptions though.)


Let $\pi:S^\prime \longrightarrow S$ be a finite morphism of Dedekind schemes and suppose that there is a morphism $P:S^\prime\longrightarrow X$ of $S$-schemes, i.e., we have that $\pi = f\circ P$.


**Question.** Does the morphism $f:X\longrightarrow S$ factor through a flat projective morphism $X\longrightarrow S^\prime$ ? 


Of course, this is not going to be true in general. So here's a better question.


**Less precise question.** If this is not true, can we ''explain'' when $f$ factors through $\pi$ and when not?


**Unprecise question.** Does a situation like the one above arise ''often''? Any references that might help?



Sections of morphisms of schemes up to a finite morphism
Consider the Stein factorization $X\to S\_1\to S$ of $f$ (thus $S\_1=\mathrm{Spec}(f\_\*\mathcal{O}\_X)$). Then $S\_1\to S$ is finite, and since $X$ is normal, I think $S\_1$ must be the normalization of $S$ in $K(X)$. Then $f$ factors through $S'$ if and only if $S\_1\to S$ does, which in turn is equivalent to Karl's condition $K(S')\subset K(X)$.


