ag.algebraic geometry – Reduction theory of higher dimensional algebraic varieties

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 $mathcal Xtooperatorname{spec} O_K$, such that $mathcal X_0cong X$ 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