# 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?

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.
• @mqx No. Blow-ups do not commute with any base-change. Suppose that you blow-up a point $p$ in $X$. What would the base-change along $p \to X$ be? – Ariyan Javanpeykar Mar 10 '16 at 8:27