Recently I was reading Moriwaki's work about the heights on arithmetic varieties, and I came across the following doubt about the setting.
Let $(X,L)$ be a couple where $X$ is a projective nonsingular algebraic variety over a number field $K$ and let $L$ a is line bundle on $X$. Then a model for $(X,L)$ is a couple $(\mathcal X,\mathcal L)$ such that $\mathcal X$ is a model over $O_K$ for $X$, and $\mathcal L$ is a $\mathbb Q$-line bundle on $\mathcal X $ such that $\mathcal L_{|K}\cong L$ (over $\mathbb Q$).
Now, I can understand the existence and the construction of $\mathcal X$, but I have some problems about $\mathcal L$:
- Why does such a $\mathbb Q$-line bundle exist?
- Why do we have to appeal to $\mathbb Q$-line bundles? I mean, why cannot $\mathcal L$ be just an ordinary line bundle on $\mathcal X$?
Essentially we are saying that there exists a line bundle $\mathcal M$ on $\mathcal X$ such that $\mathcal M_{|K}\cong n L$ for some natural number $n$. I am asking which such $\mathcal M$ exists and why it doesn't always exist with $n=1$.