On the cancellation problem for algebraic tori
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, p. 2621-2640
We address a variant of Zariski Cancellation Problem, asking whether two varieties which become isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property is easily checked for curves, is known to hold for smooth varieties of log-general type by virtue of a result of Iitaka-Fujita and more generally for non 𝔸 * 1 -uniruled varieties. We show in contrast that for smooth affine factorial 𝔸 * 1 -ruled varieties, cancellation fails in any dimension bigger than or equal to two.
Nous considérons une variante du Problème de Simplification de Zariski pour les tores algébriques : deux variétés algébriques dont les produits cartésiens avec un même tore algébrique sont isomorphes sont-elles isomorphes ? Un argument élémentaire montre que les courbes algébriques possèdent cette propriété de simplification. Un résultat très général de simplification du à Iitaka et Fujita implique qu’il en est de même pour les variétés de type log-général ou non 𝔸 * 1 -réglées. Dans cet article, nous construisons en toute dimension supérieure ou égale à deux des couples de variétés factorielles 𝔸 * 1 -réglées ne possèdant pas la propriété de simplification par des tores.
Received : 2015-06-08
Revised : 2016-02-02
Accepted : 2016-02-18
Published online : 2016-10-04
DOI : https://doi.org/10.5802/aif.3073
Classification:  14R05,  14L30
Keywords: cancellation problem, algebraic tori, principal bundles
@article{AIF_2016__66_6_2621_0,
     author = {Dubouloz, Adrien},
     title = {On the cancellation problem for algebraic tori},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {6},
     year = {2016},
     pages = {2621-2640},
     doi = {10.5802/aif.3073},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2016__66_6_2621_0}
}
On the cancellation problem for algebraic tori. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2621-2640. doi : 10.5802/aif.3073. https://aif.centre-mersenne.org/item/AIF_2016__66_6_2621_0/

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