# ag.algebraic geometry – Contraction of some surfaces over a ring of algebraic integers

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 $$Xto bar{X}$$ 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

1. Horizontal: spectra of finite flat extensions of $$mathcal{O}_K$$
2. 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.

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.