Let $\mathcal{X}$ be a smooth proper model of $X$ over $O_{K,S}$, where $K$ is a number field, $X$ is a projective curve over $K$, $O_{K,S}$ is the ring corresponding to the set $S$ of bad reductions of $X$. Let $P,Q$ be two distinct $K$-rational points of $X$, and $v\not\in S$, and $p_v$ is the associated prime ideal. Arithmetic intersection number $(P\cdot Q)_v$ is the largest positive integer $m$ such that $P$ and $Q$ have the same image in $\mathcal{X} (O_{K,S}/p_v^m)$.
I don't understand this definition. I know that if we have a projective model then $K$ points of $X$ correspond to $O_{K,S}$ points of $\mathcal{X}$. What can be said about proper models? Why shall such $m$ exist?
The fact about correspondance follows from valuative criterion of properness. For the other fact, we get it from the usual definition of the intersection numbers, for $v\in\mathrm{Spec} O_{K,S}$ we normally define $i_v(D,E)$ as sum of $i_x(D,E)[k(x):k(v)]$ over all $x$ in $\mathcal{X}$ lying above $v$, where $D,E$ two divisors. We observe that there exists only one such $x$, and that $[k(x):k(v)]=1$. And here $D,E$ are taken to be images of $\mathrm{Spec} O_{S,K}$ to $\mathcal{X}$ that correspond to points $P,Q$.