# 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