no. 6
Cylindres dans les fibrations de Mori: formes du volume quintique de del Pezzo
[Cylinders in Mori Fiber Spaces : Forms of the quintic del Pezzo threefold]
Dubouloz, Adrien1 ; Kishimoto, Takashi2
1 IMB UMR5584 CNRS, Univ. Bourgogne Franche-Comté 21000 Dijon (France)
2 Department of Mathematics Faculty of Science Saitama University Saitama 338-8570 (Japan)
Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2377-2393.

Dans le contexte général de l’étude de l’existence de 𝔸 1 -cylindres ouverts dans les variétés projectives de dimension supérieure, nous considérons le cas des fibrations de Mori de dimension relative trois, dont les fibres générales sont isomorphes au volume quintique de del Pezzo, l’unique variété de Fano lisse de degré cinq et d’indice deux. Nous établissons que les espaces totaux de fibrations de Mori de ce type contiennent toujours un 𝔸 2 -cylindre relatif au-dessus de la base de la fibration. Nous donnons également une caractérisation reliant l’existence de 𝔸 3 -cylindres relatifs à l’existence de certaines droites spéciales dans la fibre générique de ces fibrations

Motivated by the general question of existence of open 𝔸 1 -cylinders in higher dimensional projective varieties, we consider the case of Mori Fiber Spaces of relative dimension three, whose general closed fibers are isomorphic to the quintic del Pezzo threefold, the smooth Fano threefold of index two and degree five. We show that the total spaces of these Mori Fiber Spaces always contain a relative 𝔸 2 -cylinder, and we characterize those admitting relative 𝔸 3 -cylinders in terms of the existence of certain special lines in their generic fiber.

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DOI : 10.5802/aif.3297
Classification : 14E30, 14J30, 14J45, 14R10, 14R25
Keywords: Volumes de Fano, Fibrations de Mori, Liens de Sarkisov, Involutions de Cremona, Cylindres
Mot clés : Fano threefolds, Mori Fiber Space, Sarkisov link, Cremona involutions, Cylinders

Dubouloz, Adrien 1 ; Kishimoto, Takashi 2

1 IMB UMR5584 CNRS, Univ. Bourgogne Franche-Comté 21000 Dijon (France)
2 Department of Mathematics Faculty of Science Saitama University Saitama 338-8570 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Cylindres dans les fibrations de {Mori:} formes du volume quintique de del {Pezzo}},
     journal = {Annales de l'Institut Fourier},
     pages = {2377--2393},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Dubouloz, Adrien; Kishimoto, Takashi. Cylindres dans les fibrations de Mori: formes du volume quintique de del Pezzo. Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2377-2393. doi : 10.5802/aif.3297. https://aif.centre-mersenne.org/articles/10.5802/aif.3297/

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