Take the 2-minute tour ×  If we have a $G$-Galois branched
covering $Y \rightarrow \mathbb{P}^1_R$ of curves over a
complete DVR, $R$ (assume $R$ is equi-characteristic $0$, and
let $t$ be $R$'s parametrizing element). Assume that this cover
branches only at horizontal divisors (meaning it doesn't branch
along $t$ - the closed fiber). Is it possible to blow
$\mathbb{P}^1_R$ however many times (call the resulting
$R$-curve $X'$), such that the map $Y' \rightarrow X'$, where
$Y'$ is the normalization of $X'$ in the function field of $Y$,
is branched (also) along some irreducible curve of its closed
fiber (meaning it branches along some vertical divisor)?  If so,
what would be an example?  share|improve this question I love
your use of 'suddenly' :P –  Mariano Suárez-Alvarez Jul 12 '10
at 21:45 add comment  1 Answer  Yes, blowing-up $\mathbb P^1_R$
along a point of the branch locus of the special fiber of $Y\to
\mathbb P^1_R$ will give you an example. More concretely,
consider the cover $y^2=x$ of $\mathbb P^1_R