# Definition of Model of an Algebraic Curve

I'm reading the classical paper of Arakelov "Intersection Theory of Divisors on an Arithmetic Surface". At the very beginning he uses the notion of model of a curve.

In specific we have a number field $K$ with ring of integers $R$, $X$ is a smooth complete algebraic curve over $K$. Here Arakelov writes: "Let $V$ be any smooth and complete model of $X$ over $R$."

What does it mean? I didn't find the definition nor googling it, neither looking on the books "Algebraic Geometry" of Hartshorne and "Introduction to Intersection Theory in Algebraic Geometry" of Fulton.

• See Liu's "Algebraic Geometry and Arithmetic Curves", Chapter 10.
– KCd
Oct 14, 2011 at 13:37
• Thank you very much, that´s perfect! Oct 14, 2011 at 14:13

According to Definition $1.1$ of Liu´s Algebraic Geometry and Arithmetic Curves, Chapter 10 paragraph 10.1.1:
Let $S$ be a Dedekind scheme of dimension $1$, with function field $K$. Let $C$ be a normal, connected, projective curve over $K$. We call a normal fibered surface $\mathcal{C} \rightarrow S$ together with an isomorphism $f:\mathcal{C}_\eta \simeq C$ a model of $C$ over $S$. We will say that and a regular model of $C$ if $\mathcal{C}$ is regular. More generally, we will say that a model $(\mathcal{C},f)$ verifies a property $(P)$ if $\mathcal{C} \rightarrow S$ verifies $(P)$. The property $(P)$ can, for example, be the fact of being smooth, minimal regular, or regular with normal crossings, etc. A morphism $\mathcal{C} \rightarrow \mathcal{C}^´$ of two models of $C$ is a morphism of $S$-schemes that is compatible with the isomorphisms $\mathcal{C}_\eta \simeq C$, $\mathcal{C}^´_\eta \simeq C$.
• I think there is a typo here. You meant $\mathcal{C}_{\eta}$ in the second line. Otherwise I think this does not make sense. May 11, 2017 at 6:39
• @Bombyxmori old post, but I think it's correct as written: the curve over $S$ is denoted with a calligraphic $\mathcal C$ (which happens to look very similar to the original curve $C$), and $\mathcal C_{\eta}$ is its generic fiber. Sep 30, 2022 at 3:10