**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 $X$:


1. $X$ is integral.
2. $X\_K$ is geometrically integral.
3. $X \to \textrm{spec}(\mathcal{O}\_K)$ is surjective.


Let $X\to \bar{X}$ 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


1. Horizontal: spectra of finite flat extensions of $\mathcal{O}\_K$
2. Vertical: curves over $\mathcal{O}\_K/p\mathcal{O}\_K$, which is a finite field, for nonzero ideals $p\in \textrm{spec}(\mathcal{O}\_K)$.


**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, $\bar{X}\backslash V\to Y$ 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](http://%3Chttps://www.jstor.org/stable/2372985%3E) of Artin, but I couldn't understand how. I also found a [paper](http://www.numdam.org/item/10.24033/asens.1581.pdf) 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](http://www.numdam.org/item/PMIHES_1969__36__195_0.pdf) , so you may just help me to find why these conditions apply to my case.



Contraction of some surfaces over a ring of algebraic integers
The paper [2] is a seminar talk announcing the results of [3]. In this talk, I explain how to deduce the existence of integral points (theorem 1) from the contraction theorem (theorem 2). In turn, the contraction theorem is due to Artin [1] in the geometric case (surfaces over finite fields), but Raynaud explained me the arithmetic analogue (see remark 4.5/2 in [2]). Thus, the arithmetic case is *not* in Artin's paper.


Then Szpiro and I found it more convenient to prove the existence of integral points directly (and then deduce the contraction theorem): this is what I did in [3], redirecting Raynaud's arguments for the contraction theorem, especially the crucial proposition 3.8 of [3], for which I must confess that Raynaud does not get proper credit.


[1] M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485-496. doi:10.2307/2372985  

[2] L. Moret-Bailly, Points entiers des variétés arithmétiques, Séminaire de Théorie des Nombres, Paris 1985–86, p. 147-154, Progress in Mathematics vol. 71 (Birkhäuser)  

[3] L. Moret-Bailly, Groupes de Picard et problèmes de Skolem I, Ann. scient. Ec. Norm. Sup. 22 (1989), 161–179. doi: 10.24033/asens.1581


