Author: Eiko

Time: 2025-02-23 12:30:16 - 2025-02-23 12:30:16 (UTC)

References: Liu2002 Algebraic Geometry and Arithmetic Curves.

The goal is to understand Prop-10.1.21 in Liu2002 and related concepts.

Reduction Of Algebraic Curves

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 $\mathfrak{p}$,

  $$V_{\mathfrak{p}}=V{\times }_{\mathrm{Spec}({\mathcal{O}}_K)}\mathrm{Spec}({\mathcal{O}}_K/\mathfrak{p}).$$

Models Of Algebraic Curves

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 $f:\mathcal{C}\to {\mathcal{C}}^{\prime }$ 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 ${\mathcal{C}}_{\eta }^{\prime }\cong C$.