**The situation:**

Let $X$ be a 2 dimensional normal quasi-projective $mathcal{O}_K$-scheme, where $K$ is an algebraic number field. Assume the following conditions on $X$:

- $X$ is integral.
- $X_K$ is geometrically integral.
- $X to textrm{spec}(mathcal{O}_K)$ is surjective.

Let $Xto bar{X}$ an open immersion into a projective scheme. (this exist since $X$ is quasi-projective).

In particular, 1-dimensional irreducible closed subschemes of $bar{X}$ are either

- Horizontal: spectra of finite flat extensions of $mathcal{O}_K$
- Vertical: curves over $mathcal{O}_K/pmathcal{O}_K$, which is a finite field, for nonzero ideals $pin textrm{spec}(mathcal{O}_K)$.

**The claim I want to prove:**

Let $V$ be the vertical part of $bar{X}backslash X$. Theorem 2 of (Points entiers des variétés arithmétiques, Moret-Bailly) states (withouth proof) that there exists a “contraction” of $V$. i.e. a map $bar{X}to Y$ which is surjective, $bar{X}backslash Vto Y$ is an open immersion and the image in $V$ is a set of isolated points. I am looking for a proof of such a claim. If that helps, you may assume that $K=mathbb{Q}$. You may assume that $bar{X}$ is regular.

**What I have already found:**

The paper states states that this follows similarly from a paper of Artin, but I couldn’t understand how. I also found a paper of Moret-Bailly which uses the existance of integral points on $X$ to prove what I am surching for. But I am looking for a proof which does not rely on this fact, since I am trying to write the proof of theorem 1 of (Points entiers des variétés arithmétiques, Moret-Bailly) using theorem 2 of the same paper. Conditions of the existance of such a contraction can also be found in theorem 27.1 of this work , so you may just help me to find why these conditions apply to my case.