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Let $K$ and $K'$ be number fields $K \subset K'$, and let $R$ and $R'$ be the corresponding ring of integers. Let $S = Spec\ R$ and $S' = Spec\ R'$. Suppose $X \to S$ be an arithmetic surface that is a regular scheme of dimension 2 which is projective and flat over $S$.

Is $X' = X \times_S S'$ also regular? If not what kind of singularities may arise? What happens more generally?

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    $\begingroup$ The base change need not be regular. For instance, inside $\text{Proj}\ R[t,u,v]$, and for $a\in R\setminus\{0\}$, define $X_a$ to be the closed subscheme with defining ideal $\langle tv-au^2\rangle$. This is regular if and only if the valuation of $a$ with respect to each nonzero prime is $0$ or $1$. Even if this is true for valuations of $K$, it can easily fail for $K'$. $\endgroup$
    – Jason Starr
    Commented Jan 6, 2017 at 11:43
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    $\begingroup$ As both the example of Kevin Buzzard and my example make clear, if the critical locus of the morphism $X\to S$ (the closed complement of the maximal open subscheme over which the morphism is smooth) is disjoint from the ramification locus of $S'\to S$, then the base change $X'\to S'$ is regular. Without specifying more about the critical points and the ramification, it is difficult to say what type of singularities will occur, apart from the fact that they will be hypersurface singularities. $\endgroup$
    – Jason Starr
    Commented Jan 6, 2017 at 12:31
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    $\begingroup$ The basic issue is that regularity is not a relative notion (it is a miracle that for schemes locally of finite type over a perfect field $k$, it coincides with the relative notion of smoothness over the base field, and so in those cases is well-behaved under base change). You could pose the same question about normality, and again away from where the structure map or the base change are smooth (or etale) there will be big problems in general. As Jason Starr notes, semistable curves provide interesting examples (though for which normality is preserved under such Dedekind base change). $\endgroup$
    – nfdc23
    Commented Jan 6, 2017 at 18:13

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I don't think this will be true in general.

Say $K=\mathbf{Q}$ and $K'=\mathbf{Q}(\sqrt{2})$, and let $X_0$ be $Spec(R')$. Then $X_0$ is regular of dimension 1 and the map down to $S$ is projective and flat, but the base change to $S'$ is the spectrum of $\mathbf{Z}[\sqrt{2}][X]/(X^2-2)$ which is two smooth affine curves meeting in a nasty way at the point in characteristic 2. Now let $X$ be $X_0\times\mathbf{P}^1$.

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