On Albanese torsors and the elementary obstruction

Let k be a field of characteristic zero and k an algebraic closure of k. For a geometrically integral variety X over k, we write k(X) for the function field of X = X × k k. If X has a smooth k-point, the natural embedding of multiplicative groups k * ֒→ k(X) * admits a Galois-equivariant retraction. In the first part of the paper, over local and then over global fields, equivalent conditions to the existence of such a retraction are given. They are expressed in terms of the Brauer group of X.In the second part of the paper, we restrict attention to varieties which are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For k local or global, for such a variety X, in many situations but not all, the existence of a Galois-equivariant retraction to k * ֒→ k(X) * ensures the existence of a k-rational point on X. For homogeneous spaces of linear algebraic groups, the technique also handles the case where k is the function field of a complex surface. RésuméSoient k un corps de caractéristique nulle et k une clôture algébrique de k. Pour une k-variété X géométriquement intègre, on note k(X) le corps des fonctions de X = X × k k. Si X possède un k-point lisse, le plongement naturel de groupes multiplicatifs k * ֒→ k(X) * admet une rétractionéquivariante pour l'action du groupe de Galois de k sur k.Dans la première partie de l'article, sur les corps locaux puis sur les corps globaux, on donne des conditionséquivalentesà l'existence d'une telle rétractionéquivariante. Ces conditions s'expriment en terme du groupe de Brauer de la variété X.Dans la seconde partie de l'article, on considère le cas des espaces homogènes de groupes algébriques connexes, non nécessairement linéaires, avec groupes d'isotropie géométriques connexes. Pour k local ou global, pour un tel espace homogène X, 1 dans beaucoup de cas mais pas dans tous, l'existence d'une rétractionéquivariantè a k * ֒→ k(X) * implique l'existence d'un point k-rationnel sur X. Pour les espaces homogènes de groupes linéaires, la technique permet aussi de traiter le cas où k est un corps de fonctions de deux variables sur les complexes.

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“…QED Other cases where Question (1) can be answered positively will be handled in Subsections 2.2. and 2.3. For further results, see [49].…”

Section: Proof (I) Is Obviousmentioning

confidence: 99%

The elementary obstruction and homogeneous spaces

2008

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“…Let Y be a smooth projective variety over a field L. There exists an abelian variety Alb 0 (Y ) and a torsor Alb 1 (Y ) over Alb 0 (Y ) satisfying the universal property for morphisms of Y to torsors over an abelian variety [23], [25]. Let f : X → S be a projective morphism of irreducible varieties.…”

Section: Examples Of Cremona Special Setsmentioning

confidence: 99%

Cremona special sets of points in products of projective spaces

Dolgachev

2011

Complex and Differential Geometry

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A set of points in the projective plane is said to be Cremona special if its orbit with respect to the Cremona group of birational transformations consists of finitely many orbits of the projective group. This notion was extended by A. Coble to sets of points in higher-dimensional projective spaces and by S. Mukai to sets of points in the product of projective spaces. No classification of such sets is known in these cases. In the present article we survey Coble's examples of Cremona special points in projective spaces and initiate a search for new examples in the case of products of projective spaces. We also extend to the new setting the classical notion of associated points sets.

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“…This means that there exist a K -rational map X Y . By [Wittenberg 2008, Lemma 3.1.2], if we have a K -rational map X Y between smooth geometrically integral K -varieties, then ob(X ) = 0 implies ob(Y ) = 0. Since T is a K -torus, if ob(Y ) = 0, then Y (K ) = ∅; see Section 2.1 above.…”

Section: The Elementary Obstructionmentioning

confidence: 99%

“…The developments in Chapter 3 are largely motivated and inspired by work of Yekutieli and Zhang [2004;2008;2009] (see also [Yekutieli 2010]). One of their goals was to construct a new foundation for Grothendieck duality theory.…”

Section: Classification Of Semisimple Toric-friendly Groupsmentioning

confidence: 99%