The short question is: Say $p:\bar{X}\rightarrow S$ is a proper and normal morphism with the following properties:

- S is integral and smooth over a certain base field $k$,
- $\bar{X}$ has a smooth and dense open subscheme $X$, which is a smooth affine curve over S,
- $\bar{X}_s$ is of dimension $1$ on every fibre $\kappa(s)\rightarrow S$.

Then according to the literature, every integral closed subscheme $Z$ of $\bar{X}$ of codimension $1$ which is supported on $X$ should be finite and surjective over $S$. But why?

This is an assertion without proof in Voevodsky and Suslin's paper *Singular homology of abstract algebraic varieties*, Theorem 3.1. I can see that this is true for the bundle case, that is, $X=\mathbb{A}_S^1\rightarrow\mathbb{P}_S^1=\bar{X}$. In general, the closed subscheme $\bar{X}\backslash X$ should be non-empty on every fibre of $S$ since $X\rightarrow S$ is affine, so the problem seems related to something quite intuitive that

- For a proper morphism $\bar{X}\rightarrow S$ smooth of relative dimension $1$, do two elementary Weil divisors of $\bar{X}$ meet when one is over the generic point of $S$ and the other is over a elementary Weil divisor of $S$?