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Rationality problem for algebraic tori Hoshi, Akinari ; Yamasaki, Aiichi | American Mathematical Society 2017

Ouvrage

V

- v; 215 p.
ISBN 978-1-4704-2409-1

Memoirs of the american mathematical society , 1176

Localisation : Collection 1er étage

problème de rationalité # tori algébrique # résolution flasque # théorème de Krull-Schmidt # groupe de Bravais # cohomologie de Tate

11E72 ; 12F20 ; 13A50 ; 14E08 ; 20C10 ; 20G15

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Cohomological and geometric approaches to rationality problems.
New perspectives
Bogomolov, Fedor ; Tschinkel, Yuri | Birkhäuser 2010

Ouvrage

V

- ix; 311 p.
ISBN 978-0-8176-4933-3

Progress in mathematics , 0282

Localisation : Collection 1er étage

rationalité # cohomologie # géométrie algébrique # invariant cohomologique # groupe fini # groupe de Lie # espace de modules # point rationel # variétés algébriques

11R32 ; 12F12 ; 13A50 ; 14D20 ; 14E05 ; 14E08 ; 14F20 ; 14G05 ; 14G15 ; 14H10 ; 14H45 ; 14H60 ; 14J32 ; 14J35 ; 14L30

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Brauer groups and obstruction problems:
moduli spaces and arithmetic
Auel, Asher ; Hassett, Brendan ; Varilly-Alvarado, Anthony ; Viray, Bianca | Birkhäuser 2017

Ouvrage

V

- ix; 247 p.
ISBN 978-3-319-46851-8

Progress in mathematics , 0320

Localisation : Collection 1er étage

théorie des groupes # géométrie algébrique # groupe de Brauer # catégorie dérivée des faisceaux cohérents # isogénie # point de torsion # courbe modulaire # surface K3

14F05 ; 14F22 ; 14E08 ; 14G05 ; 14J28 ; 14J35 ; 14J60

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Birationally rigid varieties Pukhlikov, Aleksandr | American Mathematical Society 2013

Ouvrage

V

- vi; 367 p.
ISBN 978-0-8218-9476-7

Mathematical surveys and monographs , 0190

Localisation : Collection 1er étage

géométrie algébrique # application birationnelle # questions de rationalité # variétés algébriques spéciales # automorphisme birationnel

14E05 ; 14E07 ; 14J45 ; 14E08 ; 14E30 ; 14M10 ; 14M20 ; 14J30 ; 14J40 ; 14-02 ; 14Mxx

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Birationally rigid Fano threefold hypersurfaces Cheltsov, Ivan ; Park, Jihun | American Mathematical Society 2016

Ouvrage

y

- v; 117 p.
ISBN 978-1-4704-2316-2

Memoirs of the american mathematical society , 1167

Localisation : Collection 1er étage

hypersurface de Fano # espace projectif pondéré # rigidité birationnelle # involution birationnelle

14E07 ; 14E08 ; 14J30 ; 14J45 ; 14J70

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Stable rationality - Lecture 3 Pirutka, Alena | CIRM H

Multi angle

y

Research talks

Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational nonrational varieties. This problem remained open till 1970th, when three types of such examples were produced: cubic threefolds (Clemens and Griffiths), some quartic threefolds (Iskovskikh and Manin), and some conic bundles (Artin et Mumford). The last examples are even not stably rational. The stable rationality of the first two examples was not known.
In a recent work C. Voisin established that a double solid ramified along a very general quartic is not stably rational. Inspired by this work, we showed that many quartic solids are not stably rational (joint work with J.-L. Colliot-Thélène). More generally, B. Totaro showed that a very general hypersurface of degree d is not stably rational if d/2 is at least the smallest integer not smaller than (n+2)/3. The same method allowed us to show that the rationality is not a deformation invariant (joint with B. Hassett and Y. Tschinkel).
In this series of lectures, we will discuss the methods to obtain the results above: the universal properties of the Chow group of zero-cycles, the decomposition of the diagonal, and the specialization arguments.
Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational ...

14C15 ; 14C25 ; 14E08 ; 14H05 ; 14J70 ; 14M20

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On examples of varieties that are not stably rational Pirutka, Alena | CIRM H

Multi angle

y

Research Talks

A variety X is stably rational if a product of X and some projective space is rational. There exists examples of stably rational non rational complex varieties. In this talk we will discuss recent series of examples of varieties, which are not stably rational and not even retract rational. The proofs involve studying the properties of Chow groups of zero-cycles and the diagonal decomposition. As concrete examples, we will discuss some quartic double solids (C. Voisin), quartic threefolds (a joint work with Colliot-Thélène), some hypersurfaces (Totaro) and others. A variety X is stably rational if a product of X and some projective space is rational. There exists examples of stably rational non rational complex varieties. In this talk we will discuss recent series of examples of varieties, which are not stably rational and not even retract rational. The proofs involve studying the properties of Chow groups of zero-cycles and the diagonal decomposition. As concrete examples, we will discuss some quartic ...

14C15 ; 14M20 ; 14E08

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