References: Liu2002 Algebraic Geometry and Arithmetic Curves.
The goal is to understand Prop-10.1.21 in Liu2002 and related concepts.
To study the arithmetic of an variety \(V/K\), it is necessary to look at its behaviour modulo finite and infinite places of \(K\). In reduction, we consider all the finite places. This involves:
Extend \(V/K\) to a scheme over integers \(V/\mathcal{O}_K\) and try to preserve as much good stuff as possible.
The reduction of \(V\) mod maximal ideal \(\mathfrak{p}\) is just the fibre of \(V\to \mathrm{Spec}(\mathcal{O}_K)\) over \(\mathfrak{p}\),
\[V_\mathfrak{p}= V\times_{\mathrm{Spec}(\mathcal{O}_K)}\mathrm{Spec}(\mathcal{O}_K/\mathfrak{p}).\]
Let \(S\) be Dedekind scheme (a Noetherian integral scheme of dimension \(1\) all of its local rings are regular / normal). A Dedekind scheme has at least \(2\) points, one generic point and the rest are all closed points. We use \(K\) to denote its function field.
Let \(C\) be a normal connected projective curve over \(K\) (think of it as defined generically on \(S\)).
Definition.
A model of \(C/S\) is a normal fibered surface \(\mathcal{C}\to S\) with an isomorphism \(\mathcal{C}_\eta\cong C\).
Such model is regular if \(\mathcal{C}\) is regular.
We say a model \((\mathcal{C}, f)\) satisfy \(P\) if \(\mathcal{C}\to S\) satisfy \(P\). (e.g. \(P=\) smooth, minimal regular, regular with normal crossings, etc.)
A morphism of models of \(f:\mathcal{C}\to \mathcal{C}'\) is a morphism of schemes over \(S\) and also over \(C\), i.e. compatible with the isomorphism on generic fibers \(\mathcal{C}_\eta\cong C\) and \(\mathcal{C}'_\eta\cong C\).