## Néron models.(English)Zbl 0705.14001

Let $$S$$ be a connected Dedekind scheme with field of rational functions $$K$$ and let $$X_ K$$ be a smooth and separated $$K$$-scheme of finite type. A Néron model of $$X_ K$$ is a smooth separated $$S$$-scheme $$X$$ of finite type with generic fibre $$X_ K$$ satisfying the following universal property: for each smooth $$S$$-scheme $$Y$$ and each $$K$$-morphism $$u_ K: Y_ K\to X_ K$$ there exists a unique $$S$$-morphism $$u: Y\to X$$ extending $$u_ k$$.
This book is devoted to the construction of the Néron models and to the study of their properties. In particular, in the case of a relative curve $$X\to S$$, the Néron model of the Jacobian $$J_ K$$ of the generic fibre $$X_ K$$ is studied and its relationship with the relative Picard functor is explained. The book contains also an ample exposition of the main tools and methods used so that, for example, chapter 8 and chapter 9 are a useful reference for questions related to the Picard functor and to its representability.

### MSC:

 14A15 Schemes and morphisms 14K30 Picard schemes, higher Jacobians 14L15 Group schemes 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14C22 Picard groups

### Keywords:

Néron model of the Jacobian; relative Picard functor