Néron models were invented by A. Néron in the early 1960s in order to study the integral structure of abelian varieties over number fields. Since then, arithmeticians and algebraic geometers have applied the theory of Néron models with great success. Quite recently, new developments in arithmetic algebraic geometry have prompted a desire to understand more about Néron models, and even to go back to the basics of their construction. The authors have taken this as their incentive to present a comprehensive treatment of Néron models. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of Néron models from the point of view of Grothendieck's algebraic geometry. In the second part of the book the relationship between Néron models and the relative Picard functor in the case of Jacobian varieties is explained. The authors helpfully remind the reader of some important standard techniques of algebraic geometry. A special chapter surveys the theory of the Picard functor.
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Some Background Material from Algebraic Geometry
The Smoothening Process
8 other sections not shown
abelian variety admits a Neron affine open algebraic space assertion assume base change bijective canonical map canonical morphism Cartier divisor closed immersion closed point closed subscheme coincides commutative condition consider construction Corollary curve Dedekind scheme defined denote descent datum diagram discrete valuation ring element etale etale morphisms exact sequence exists extension faithfully flat field of fractions finite presentation finite type follows Furthermore geometric gives rise global sections group law group scheme hence henselian induced invertible sheaf irreducible components isomorphism K-group Lemma line bundle locally of finite multiplication Neron lft-model Neron model noetherian open immersion Picard functor Picx/s polynomial projective Proof proper Proposition pull-back quasi-compact quotient R-model rational map reduced representable residue field resp S-birational S-morphism satisfies Sch/S schematic closure smooth and separated smooth locus smoothening Spec special fibre Xk strictly henselian subgroup subset surjective Theorem topology torsor unipotent