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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?

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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 $\mathbb P^1_R$ by its self. When you blow-up the point $(x=t=0)$, you get an open subset $y^2=tu$ in $Y'$ ($u=x/t$). The exceptional divisor $E$ in $X'$ is in the branch locus because $E$ has multiplicity $1$ in the special fiber of $X'$, while its preimage in $Y'$ has multiplicity $2$.

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