By arithmetic surface we mean a projective, regular and flat scheme of dimension 1 over $Spec(O_K)$ for some algebraic number field $K$.
So we can view such a scheme $X$ as a closed subscheme of $Proj(O_K[T_0,...,T_n])$ or equivalently $Proj(O_K[T_0,...,T_n]/I)$ for some homogeneous ideal $I$.
So what can we get for $I$ from the conditions that $Proj(O_K[T_0,...,T_n]/I)\rightarrow Spec(O_K)$ is regular, flat and of dimension 1?
