Let $:f:X\to Y$ be a projective surjective morphism between two normal varieties over $\mathbb{C}$. Assume that $f$ has only $1$-dimensional fibers. Let $D$ be a multi-section of $f$, i.e., $D$ is a prime Weil divisor on $X$ such that $f|\_D: D\to Y$ is a generically finite surjective morphism. Let $D\to Y''\to Y $ be the Stein factorization of $f|\_D$ and $Y'\to Y''$ the normalization morphism. Now we do a fiber product between $f:X\to $Y and $\phi:Y'\to Y$, i.e., $X\times\_Y Y'$.
Let $X'$ be the normalization of the main component of $X\times\_Y Y'$ dominating $Y'$. Let $f':X'\to Y'$ and $\phi':X'\to X$ be the corresponding projections. 


Now in this scenario, I have seen in several papers so far now (without any explanation) that **$\phi'^{-1}D$ splits into irreducible components, at least one of which becomes a section for $f':X'\to Y'$ over a dense open subset of $Y'$**.


My question is why is this true? How would one prove it formally? I actually need to make all the components of $\phi'^{-1}D$ into sections over some dense open set, so I believe I will have to perform this kind base changes few more times, then how I do know this process stops after finitely many steps?


\*\* I assumed $D$ is only a prime Weil diviso but it helps you may assume that it is also a Cartier divisor, so that pullback makes sense.


\*\* I would like to mention that this seems to be a standard technique in proving the Canonical Bundle Formula when the relative dimension is $1$. For example, this argument appears in the proof of Theorem 8.1 in the paper "Towards the second main theorem on complements" by Prokhorov and Shokurov (2009). 



How to split a Multi-section into finitely many Sections via base-change?
First, you have a multisection $D\_1 = D \times\_Y Y'$ for the family $X' \to Y'$, which is still generically finite of the same degree over $Y'$. 


On the other hand, let $D' = D \times\_{Y''} Y'$. Then the maps $D' \to D \to X$ and $D' \to Y'$ induce a map $D' \to X'$ which is generically finite over $Y'$ with connected fibers (by properties of the Stein decomposition). Hence it is generically one-to-one onto $Y'$. On the other hand, the image of $D'$ in $X'$ sits in $D\_1$. Therefore, $D\_1$ splits as the union of the image of $D'$ and an extra component, say $D\_2$. Now $D\_2$ is generically finite over $Y'$ of degree by 1 less than the degree of $D$. So, we can continue the procedure and be sure that it eventually stops.


