[Sur le Problème de Simplification pour les tores algébriques]
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 -réglées. Dans cet article, nous construisons en toute dimension supérieure ou égale à deux des couples de variétés factorielles -réglées ne possèdant pas la propriété de simplification par des tores.
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 -uniruled varieties. We show in contrast that for smooth affine factorial -ruled varieties, cancellation fails in any dimension bigger than or equal to two.
Révisé le :
Accepté le :
Publié le :
Keywords: cancellation problem, algebraic tori, principal bundles
Mot clés : problème de simplification, tores algébriques, fibrés principaux
@article{AIF_2016__66_6_2621_0,
author = {Dubouloz, Adrien},
title = {On the cancellation problem for algebraic tori},
journal = {Annales de l'Institut Fourier},
pages = {2621--2640},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {66},
number = {6},
year = {2016},
doi = {10.5802/aif.3073},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.3073/}
}
TY - JOUR AU - Dubouloz, Adrien TI - On the cancellation problem for algebraic tori JO - Annales de l'Institut Fourier PY - 2016 SP - 2621 EP - 2640 VL - 66 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3073/ DO - 10.5802/aif.3073 LA - en ID - AIF_2016__66_6_2621_0 ER -
%0 Journal Article %A Dubouloz, Adrien %T On the cancellation problem for algebraic tori %J Annales de l'Institut Fourier %D 2016 %P 2621-2640 %V 66 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3073/ %R 10.5802/aif.3073 %G en %F AIF_2016__66_6_2621_0
Dubouloz, Adrien. On the cancellation problem for algebraic tori. Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2621-2640. doi : 10.5802/aif.3073. http://www.numdam.org/articles/10.5802/aif.3073/
[1] Polynomial fibre rings of algebras over Noetherian rings, Invent. Math., Volume 87 (1987) no. 1, pp. 101-127 | DOI
[2] Uniformization of Deligne-Mumford curves, J. Reine Angew. Math., Volume 599 (2006), pp. 111-153 | DOI
[3] On a cancellation problem and automorphism groups of affine algebraic varieties (1989) (preprint Warsaw)
[4] On exotic affine 3-spheres, J. Algebraic Geom., Volume 23 (2014) no. 3, pp. 445-469 | DOI
[5] On the geometry of affine algebraic -surfaces, Problems in the theory of surfaces and their classification (Cortona, 1988) (Sympos. Math., XXXII), Academic Press, London, 1991, pp. 111-140
[6] Laurent cancellation for rings of transcendence degree one over a field, Automorphisms in birational and affine geometry (Springer Proc. Math. Stat.), Volume 79, Springer, Cham, 2014, pp. 313-326 | DOI
[7] Cohomologie non abélienne, Springer-Verlag, Berlin-New York, 1971, ix+467 pages (Die Grundlehren der mathematischen Wissenschaften, Band 179)
[8] Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. (1966) no. 28, 255 pages
[9] On the cancellation problem for the affine space in characteristic , Invent. Math., Volume 195 (2014) no. 1, pp. 279-288 | DOI
[10] A classification of factorial surfaces of nongeneral type, Michigan Math. J., Volume 61 (2012) no. 3, pp. 517-529 | DOI
[11] Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, xvi+496 pages (Graduate Texts in Mathematics, No. 52)
[12] On logarithmic Kodaira dimension of algebraic varieties, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 175-189
[13] Cancellation theorem for algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 24 (1977) no. 1, pp. 123-127
[14] Addition formula of logarithmic Kodaira dimensions for morphisms of relative dimension one, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo (1978), pp. 207-217
[15] The Picard sequences of a fibration, Proc. Amer. Math. Soc., Volume 53 (1975) no. 1, pp. 37-40
[16] Open algebraic surfaces, CRM Monograph Series, 12, American Mathematical Society, Providence, RI, 2001, viii+259 pages
[17] Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ., Volume 20 (1980) no. 1, pp. 11-42
Cité par Sources :
