Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ be the ring of integers of $F$. Assume that the residue field of $R$ is algebraically closed. Does there exist a model of $C$ over $R$ such that the special fiber is irreducible with at worst nodal singularities? 


I know that by Deligne-Mumford, there always exists a semi-stable model, which only guarentees a reducible curve with at worst nodal singularities, but I am asking if we can do better?



Model over DVR for smooth projective curves
By Lemma 1.12 in the Deligne-Mumford paper, the answer is no. Any isomorphism of generic fibers extends uniquely to an isomorphism over $R$. This is one way of saying that the stack $\mathcal{M}\_g$ of genus $g$ stable curves is proper.


