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On Zariski's theorem in positive characteristic

Ilya Tyomkin (2013)

Journal of the European Mathematical Society

In the current paper we show that the dimension of a family V of irreducible reduced curves in a given ample linear system on a toric surface S over an algebraically closed field is bounded from above by - K S . C + p g ( C ) - 1 , where C denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality 𝚍𝚒𝚖 ( V ) = - K S . C + p g ( C ) - 1 does not imply the nodality of C even if C belongs to the...

Pseudo-abelian varieties

Burt Totaro (2013)

Annales scientifiques de l'École Normale Supérieure

Chevalley’s theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field k to be a smooth connected k -group in which every smooth connected affine normal k -subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension...

Semistable reduction and torsion subgroups of abelian varieties

Alice Silverberg, Yuri G. Zarhin (1995)

Annales de l'institut Fourier

The main result of this paper implies that if an abelian variety over a field F has a maximal isotropic subgroup of n -torsion points all of which are defined over F , and n ≥ 5 , then the abelian variety has semistable reduction away from n . This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its n -torsion points are defined over a field F and n ≥ 3 , then the abelian variety has semistable reduction away from n . We also give information about the Néron models...

Sur le groupe des classes d’un schéma arithmétique

Bruno Kahn (2006)

Bulletin de la Société Mathématique de France

Nous donnons une démonstration du fait que le groupe des classes d’un schéma irréductible de type fini sur Spec 𝐙 est de type fini. Cette preuve ne repose pas sur le théorème de Mordell-Weil-Néron, mais plutôt sur le théorème de Mordell-Weil classique, le théorème de Néron-Severi et les théorèmes de Hironaka et de Jong sur la résolution des singularités. Nous en déduisons quelques corollaires, parmi lesquels le théorème de Mordell-Weil-Néron lui-même.

Variétés abéliennes et invariants arithmétiques

Jean Gillibert (2006)

Annales de l’institut Fourier

Dans la continuité de nos travaux précédents, nous étudions un analogue, pour le modèle de Néron d’une variété abélienne semi-stable sur un corps de nombres, du class-invariant homomorphism introduit par M. J. Taylor, qui nous permet de mesurer la structure galoisienne de certains torseurs.

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