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Contraction of excess fibres between the McKay correspondences in dimensions two and three

Samuel Boissière, Alessandra Sarti (2007)

Annales de l’institut Fourier

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3 surfaces appearing as minimal resolutions...

Contraction par Frobenius de G -modules

Michel Gros, Masaharu Kaneda (2011)

Annales de l’institut Fourier

Soit G un groupe algébrique semi-simple simplement connexe défini sur un corps algébriquement clos 𝕜 de caractéristique positive. Nous donnons une nouvelle preuve de l’existence d’un scindage de Frobenius de la variété des drapeaux de G ainsi que de la nature G -équivariante de celui-ci. L’outil principal est un scindage de l’endomorphisme de Frobenius défini sur toute l’algèbre des distributions de G qui permet de « détordre » la structure des G -modules.

Contractions of Lie algebras and algebraic groups

Dietrich Burde (2007)

Archivum Mathematicum

Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups.

Contractions of smooth varieties. II. Computations and applications

Marco Andreatta, Jarosław A. Wiśniewski (1998)

Bollettino dell'Unione Matematica Italiana

Una contrazione su una varietà proiettiva liscia X è data da una mappa φ : X → Z propria, suriettiva e a fibre connesse in una varietà irriducibile normale Z . La contrazione si dice di Fano-Mori se inoltre - K X è φ -ampio. Nel lavoro, naturale seguito e completamento delle ricerche introdotte in [A-W3], si studiano diversi aspetti delle contrazioni di Fano-Mori attraverso esempi (capitolo 1) e teoremi di struttura (capitoli 3 e 4). Si discutono anche alcune applicazioni allo studio di morfismi birazionali propri...

Contre-exemples au principe de Hasse pour certains tores coflasques

Régis de la Bretèche, Tim Browning (2014)

Journal de Théorie des Nombres de Bordeaux

Nous étudions le comportement asymptotique du nombre de variétés dans une certaine classe ne satisfaisant pas le principe de Hasse. Cette étude repose sur des résultats récemment obtenus par Colliot-Thélène [3].

Convex bodies associated to linear series

Robert Lazarsfeld, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens...

Co-rank and Betti number of a group

Irina Gelbukh (2015)

Czechoslovak Mathematical Journal

For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group's rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive...

Corestriction of central simple algebras and families of Mumford-type

Federica Galluzzi (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let M be a family of Mumford-type, that is, a family of polarized complex abelian fourfolds as introduced by Mumford in [9]. This family is defined starting from a quaternion algebra A over a real cubic number field and imposing a condition to the corestriction of such A . In this paper, under some extra conditions on the algebra A , we make this condition explicit and in this way we are able to describe the polarization and the complex structures of the fibers. Then, we look at the non simple C ⁢ M -fibers...

Corps de définition et points rationnels

Geoffroy Derome (2003)

Journal de théorie des nombres de Bordeaux

Soit 𝔒 un objet algébrique (par exemple une courbe ou un revêtement) défini sur ℚ ¯ et de corps des modules un corps de nombres K . Il est bien connu que 𝔒 n’admet pas nécessairement de K -modèle. En utilisant deux résultats récents dus à P. Dèbes, J.-C. Douai et M. Emsalem nous donnerons un majorant pour le degré d’un corps de définition de 𝔒 sur K . Dans une deuxième partie, nous donnerons des conditions suffisantes sur l’ordre de Aut( 𝔒 ) pour que 𝔒 admette un K -modèle.

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