Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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
1  
I love your use of 'suddenly' :P –  Mariano Suárez-Alvarez Jul 12 '10 at 21:45
add comment

1 Answer 1

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 $2$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.