Sorry for my bad English.
Let $S$ be a Dedekind scheme of dim $1$, $K$ be the fraction field, and $C$ be a normal connected projective curve over $K$.
For closed point $p\in C$, is next two notion is different?
(1)there is model $\cal{C}$ over $\operatorname{Spec} \cal{O}_{C,p}$ s.t. $\cal{C}$ is smooth over $\operatorname{Spec} \cal{O}_{C,p}$ .
(2) there is model $\cal{C}$ over $\operatorname{Spec} \cal{O}_{C,p}$ s.t. $\cal{C}\times \operatorname{Spec}k(p)$ is smooth over $\operatorname{Spec} k(p)$ .
where $k(p)$ is residue field of $p$. It's trivial that (1)$\Rightarrow $ (2) by base change. When we say $C$ is good reduction at $p$, which does it imply?
I have second question. When we say $C$ has good reduction, which means of the following?
(a)for all $p\in C$, $C$ has good reduction at $p$.
(b) there is smooth model over $S$.