# Computing stable reduction of finite covers of curves in practice

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".

### Background

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.

### Question

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?

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.