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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?

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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."

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    $\begingroup$ Is it true that blow up commute with any base change? $\endgroup$
    – user93417
    Mar 10, 2016 at 4:00
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    $\begingroup$ @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? $\endgroup$ Mar 10, 2016 at 8:27

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