# What would a arithmetic surface look like

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?