The general theory is described in various places, but I'll be following (sketchily) the description of this process appearing in section 1 of Bouw and Wewers' "Reduction of covers and Hurwitz spaces".


Let $R$ be a complete DVR, and $K$ its function field. Say we have a $G$-Galois map of (smooth projective) curves over $K$, $f:Y_K \rightarrow X_K$. Assume also that the order of $G$ is not divisible by the characteristic of the residue field of $R$. After replacing $K$ by a finite extension we may assume the ramification points are $K$-rational, and and the smooth stably marked curve $(Y_K, D)$ (where $D$ is the ramification divisor) can be defined over $R$: $(Y_R,D_R)$. There is some variation between different papers as to what "stably marked curve" means, but I think I mean minimal semi-stable, which happens to be stable (am I wrong? correct me if I am.) If we quotient $Y_R$ by the action of $G$ we should get a semi-stable curve, which we shall denote: $X_R$. This may no longer be a minimal semi-stable model of $X_K$ (but it's definitely a semi-stable model of it).

If I understand the theory correctly, if we assume that $K$ is such that we have an $R$-model of $X_K$ which is semi-stable and such that the branch points specialize to different points, then it must be $X_R$ as constructed above.


In order to understand this better, I wish to have some concrete computations under my belt. Let's try a simple yet interesting example: Let $R:= \mathbb{C}[[t]]$, $X_{\mathbb{C}((t))}:=\mathbb{P}_{\mathbb{C}((t))}^1$ (with parameter $x$), and let $f$ and $Y _ { \mathbb{C} ((t))}$ be given affinely by $y^2=x(x-t)$. (So $f$ is the projection to $x$, and $Y _ { \mathbb{C} ((t))}$ is a $\mathbb{P}_{\mathbb{C}((t))}^1$ with parameter $y/x$. In other words the function field of $X$ is $\mathbb{C}((t))(x)$ and the function field of $Y$ is $Quot(\mathbb{C}((t))[x,y]/(y^2-x(x-t)))$, which, in turn, is equal to $\mathbb{C}((t))(y/x)$.)

If we let $X_{\mathbb{C}[[t]]}:=\mathbb{P}_{\mathbb{C}[[t]]}^1$, then this is clearly a semi-stable curve, and the branch points (in $X _ { \mathbb{C}((t))}$), which were 0 and t, specialize to the same point. But I want to guarantee that this would be the quotient of the stably marked curve on top. According to the last paragraph in the background section, I would get this guarantee if the branch points (interpreted in $X_{\mathbb{C}[[t]]}$) would specialize to different points. So instead choose $X_{\mathbb{C}[[t]]}$ to be the blow up of $\mathbb{P} _ { \mathbb{C} [[t]]}^1$ at $t=x=0$. If we work affinely, this would be: $\mathbb{C}[[t]][x,z]/(xz-t)$. The question now is: how do I find the stable reduction upstairs, and the map between them? How do I finish this example?


1 Answer 1


If the normalization of $X_R$ in the function field of $Y_K$ has no vertical ramification, then I think so. You might have a look here, § 2 and 3.


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