The Stacks project

Theorem 55.18.1. Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$. Then there exists an extension of discrete valuation rings $R \subset R'$ which induces a finite separable extension of fraction fields $K'/K$ such that $C_{K'}$ has semistable reduction. More precisely, we have the following

1. If the genus of $C$ is zero, then there exists a degree $2$ separable extension $K'/K$ such that $C_{K'} \cong \mathbf{P}^1_{K'}$ and hence $C_{K'}$ is isomorphic to the generic fibre of the smooth projective scheme $\mathbf{P}^1_{R'}$ over the integral closure $R'$ of $R$ in $K'$.

2. If the genus of $C$ is one, then there exists a finite separable extension $K'/K$ such that $C_{K'}$ has semistable reduction over $R'_\mathfrak m$ for every maximal ideal $\mathfrak m$ of the integral closure $R'$ of $R$ in $K'$. Moreover, the special fibre of the (unique) minimal model of $C_{K'}$ over $R'_\mathfrak m$ is either a smooth genus one curve or a cycle of rational curves.

3. If the genus $g$ of $C$ is greater than one, then there exists a finite separable extension $K'/K$ of degree at most $B_ g$ (55.18.0.1) such that $C_{K'}$ has semistable reduction over $R'_\mathfrak m$ for every maximal ideal $\mathfrak m$ of the integral closure $R'$ of $R$ in $K'$.