A computation of ramification

Question

I tried writing the full motivation for this question, but it turned out to be too long a detour for a question that is really quite specific. So I will only sketch the motivation this time.

Sketch of motivation

I'm taking a $G$-Galois cover of $\mathbb{P}^1_{\overline{\mathbb{C}(t)}}$ which is defined (not including the group action) over $\mathbb{C}(t)$. Then I descend to $\mathbb{C}[t]$ and observe the branching behavior. The following specific example came up from a very explicit computation of a $D_8$-cover, $X$ over $\mathbb{P}^1_{\overline{\mathbb{C}(t)}}$, where the interesting part (the part in the question) is the $C_4$-cover $X \rightarrow X/C_4$. In the question, I'm actually looking formally about $t=0$.

While this is the motivation, the question will require none of this.

Question

Let $K=\mathbb{C}((t))(y)$, and $L=Quot(\mathbb{C}((t))[y,z]/z^4-(y-\sqrt{1-t})^2(y+\sqrt{1-t})^2(\sqrt{2(2-t)}-y)(\sqrt{2(2-t)}+y)^3)$. Let's look at the surface $R:=\mathbb{P}^1_{\mathbb{C}[[t]]}$ (with parameter $y$, so its function field is $K$), and the branched covering of it: $S:=$ the normalization of $R$ in $L$ (obviously with function field $L$). My question is: what is the ramification index of the vertical divisor, $t$, of $R$, in the branched covering $S \rightarrow R$?

It seems to me, through roundabout ways, that the ramification index should equal $2$, but surely there should be a systematic way of doing this that isn't completely terrible.

Do you have any tricks up your sleeves for computing vertical ramification in situations like this?

Remark

Fix a choice for $\sqrt{1-t}$ and $\sqrt{2(2-t)}$ in $\mathbb{C}[[t]]$ throughout.

Answer 1

I think there is no vertical ramification. Denote by $D={\mathbb C}[[t]]$ and $K_0$ its fraction field. Your curve is defined over $K_0$ by an affine equation $z^n+f(y)=0$ with $f(Y)\in D[Y]$ monic. Let $U$ be the affine scheme associated to $D[y,z, {{1}\over{f(y)}}]/(z^n-f(y))$. It is clearly finite and étale over the open subset $V:={\mathrm{Spec}}(D[y, {{1}\over{f(y)}}])$ of $R$, so $U=S\times_R V$. On the special fibers we have $U_s=S_s\times_{R_s} V_s$. As $V_s$ contains the generic point of $R_s$, we see that $S_s$ has a unique generic point, so $U_s$ is dense in $S_s$. But $U_s$ is étale over $V_s$, so there is no vertical ramification. What is needed here is only $D$ is a DVR and $n$ is coprime with the residue characteristic of $D$.

In fact, in your example, the zeros of $f(Y)$ remain pairwise distinct mod $t$, this implies that the function field of $U_s$ has the same genus as $L$. It is then easy to see that the whole curve $S_s$ is smooth.