Hello,

my question probably isn't too hard, but I can't find the answer.

Let $K$ be a number field, $\mathcal{X} /\mathcal{O}_K$ be an arithmetic surface, which is a regular model for a projective, smooth curve $X/K$ and let $(\mathcal{L}, ||\cdot ||)$ be a hermitian line bundle on $X$. Then every point $P\in X(K)$ admits a homomorphism $f_P : Spec \mathcal{O}_K \to \mathcal{X}$.

This is what is said and I don't understand why $f_P$ must exist and what it does.

Thanks in advance for answers!