Example 55.16.1. Let $k$ be an algebraically closed field. Let $Z$ be a smooth projective curve over $k$ of positive genus $g$. Let $n \geq 1$ be an integer prime to the characteristic of $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ Z$-module of order $n$, see Algebraic Curves, Lemma 53.17.1. Pick an isomorphism $\varphi : \mathcal{L}^{\otimes n} \to \mathcal{O}_ Z$. Set $R = k[[\pi ]]$ with fraction field $K = k((\pi ))$. Denote $Z_ R$ the base change of $Z$ to $R$. Let $\mathcal{L}_ R$ be the pullback of $\mathcal{L}$ to $Z_ R$. Consider the finite flat morphism
such that
More precisely, if $U = \mathop{\mathrm{Spec}}(A) \subset Z$ is an affine open such that $\mathcal{L}|_ U$ is trivialized by a section $s$ with $\varphi (s^{\otimes n}) = f$ (with $f$ a unit), then
The reader verifies that the morphism $X_ K \to Z_ K$ of generic fibres is finite étale. Looking at the description of the structure sheaf we see that $H^0(X, \mathcal{O}_ X) = R$ and $H^0(X_ K, \mathcal{O}_{X_ K}) = K$. By Riemann-Hurwitz (Algebraic Curves, Lemma 53.12.4) the genus of $X_ K$ is $n(g - 1) + 1$. In particular $X_ K$ has genus $1$, if $Z$ has genus $1$. On the other hand, the scheme $X$ is regular by the local equation above and the special fibre $X_ k$ is $n$ times the reduced special fibre as an effective Cartier divisor. It follows that any finite extension $K'/K$ over which $X_ K$ attains semistable reduction has to ramify with ramification index at least $n$ (some details omitted). Thus there does not exist a universal bound for the degree of an extension over which a genus $1$ curve attains semistable reduction.
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