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**Well-adjusted models for curves over Dedekind rings.**
*(English)*
Zbl 0734.14003

Arithmetic algebraic geometry, Proc. Conf., Texel/Neth. 1989, Prog. Math. 89, 3-24 (1991).

[For the entire collection see Zbl 0711.00011.]

Let \({\mathfrak O}\) be an excellent Dedekind ring and put \(Y=Spec({\mathfrak O})\). A “curve” over Y is a scheme C with a flat map \(\pi\) : \(C\to Y\) of finite type, whose fibers have dimension 1 and whose general fiber is smooth and geometrically irreducible. C need not be regular or complete.

The authors collect in this paper two results on minimal models of C. First they prove the existence of minimal elements in the class of regular curves \(C'\) which have a proper birational Y-morphism \(C'\to C\); the minimal element is in fact unique, up to isomorphism over C.

Next, the authors provide, through a scheme-theoretical construction, a new proof of a theorem of M. Artin: if C is complete, there is a finite extension L of the fraction field of \({\mathfrak O}\) such that a minimal model of \(C_ L\) is reduced over the closed point, with non-singular and geometrically irreducible components.

Let \({\mathfrak O}\) be an excellent Dedekind ring and put \(Y=Spec({\mathfrak O})\). A “curve” over Y is a scheme C with a flat map \(\pi\) : \(C\to Y\) of finite type, whose fibers have dimension 1 and whose general fiber is smooth and geometrically irreducible. C need not be regular or complete.

The authors collect in this paper two results on minimal models of C. First they prove the existence of minimal elements in the class of regular curves \(C'\) which have a proper birational Y-morphism \(C'\to C\); the minimal element is in fact unique, up to isomorphism over C.

Next, the authors provide, through a scheme-theoretical construction, a new proof of a theorem of M. Artin: if C is complete, there is a finite extension L of the fraction field of \({\mathfrak O}\) such that a minimal model of \(C_ L\) is reduced over the closed point, with non-singular and geometrically irreducible components.

Reviewer: L.Chiantini (Napoli)

### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

14H10 | Families, moduli of curves (algebraic) |

13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |

14G27 | Other nonalgebraically closed ground fields in algebraic geometry |