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?

  • $\begingroup$ No. If you have a model where the special fiber is stable and reducible (for instance, the union of 2 smooth curves meeting transversally, both of genus $>0$), you cannot find a semi-stable model with an irreducible fiber. $\endgroup$ – abx Jul 2 '15 at 20:51

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.

  • $\begingroup$ note that the uniqueness requires stability, which includes a condition on not having certain irreducible components. However I guess having one irreducible component always implies this $\endgroup$ – Will Sawin Jul 4 '15 at 1:05

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