Cylindres dans les fibrations de Mori: formes du volume quintique de del Pezzo

Dubouloz, Adrien; Kishimoto, Takashi

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.

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

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