Does blow up of subscheme in special fiber change the generic fiber? Let $X\to \mathrm{Spec}(R)$ be a finite type scheme over DVR, choose a closed subscheme $Y$ of the closed fiber $X_0$ and blow up $Y$ in $X$, will the generic fiber always remain the same?

Note that passing to the generic fiber is flat, (it's a localization) and blowups commute with flat base change.
Therefore, the new generic fiber is the blowup of the old generic fiber $X_\eta$ at the intersection of $X_\eta$ with $Y$. This is empty, so you're blowing up nothing, and you get an isomorphism.
This is just a long-winded way of saying "when you blow up a subscheme, the compliment of that closed set never ever changes."
