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If $X$ is a nonsigular curve over a number field $K$, one can obtain several arithmetic models of $X$. Namely, we can construct an arithmetic surface $\mathcal X\to\operatorname{spec} O_K$, such that $\mathcal X_0\cong X$ and with certain properties: minimality, regularity etc etc. This theory is well known an beautifully explained in Liu's book (Chapter 10). Then the arithmetic properties of $X$ are reflected on the fibres $\mathcal X_{\mathfrak p}$.

Is this theory well developed also when $X$ is a variety of dimension $d>1$? So far I have seen papers treating just very special cases: K3 surfaces, del Pezzo...

What is the theoretical obstruction to have a general theory like in the case of curves? At least I would expect that something can be said about principally polarized varieties of general type

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    $\begingroup$ A closely related problem is finding minimal regular models for singular algebraic varieties. The relation is direct in the case when $K$ is an equal characteristic local field, as then schemes over $\mathcal O_K$ can be spread out to families of varieties over a curve, and usually the mixed characteristic local field is harder than the equal characteristic case. The existence of any regular model at all is unknown in characteristic $p$, but known for surfaces (i.e. curves over a curve). $\endgroup$
    – Will Sawin
    Commented Jan 31, 2021 at 16:01
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    $\begingroup$ Even in characteristic $0$, a good theory of a minimal model does not exist in general, and when it does, it is not necessarily regular (the "minimal model program" instead uses other, less restrictive, conditions on the singularities). The situation is even less clear in characteristic $p$ and less clear than that in mixed characteristic. $\endgroup$
    – Will Sawin
    Commented Jan 31, 2021 at 16:02

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