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The answer is no (even in case 2). Typically if you take a proper and smooth relative surface $X$ over $\mathbb{Z}_{(p)}$, and you blow up a point at some point of the special fiber of $X$, the resulting scheme $Y$ will still be proper and smooth with same generic fiber, but the special fiber may have changed. Edit: Oops, Ulrich is write, the example is incorrect. Sorry. |
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The answer is no (even in case 2). Typically if you take a proper and smooth relative surface $X$ over $\mathbb{Z}_{(p)}$, and you blow up a point at some point of the special fiber of $X$, the resulting scheme $Y$ will still be proper and smooth with same generic fiber, but the special fiber may have changed. |
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