Cylindres dans les fibrations de Mori: formes du volume quintique de del Pezzo
Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2377-2393.

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.

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

Received:
Revised:
Accepted:
Published online:
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)
@article{AIF_2019__69_6_2377_0,
     author = {Dubouloz, Adrien and Kishimoto, Takashi},
     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},
     volume = {69},
     number = {6},
     year = {2019},
     doi = {10.5802/aif.3297},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.3297/}
}
TY  - JOUR
AU  - Dubouloz, Adrien
AU  - Kishimoto, Takashi
TI  - Cylindres dans les fibrations de Mori: formes du volume quintique de del Pezzo
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 2377
EP  - 2393
VL  - 69
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.3297/
DO  - 10.5802/aif.3297
LA  - en
ID  - AIF_2019__69_6_2377_0
ER  - 
%0 Journal Article
%A Dubouloz, Adrien
%A Kishimoto, Takashi
%T Cylindres dans les fibrations de Mori: formes du volume quintique de del Pezzo
%J Annales de l'Institut Fourier
%D 2019
%P 2377-2393
%V 69
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.3297/
%R 10.5802/aif.3297
%G en
%F AIF_2019__69_6_2377_0
Dubouloz, Adrien; Kishimoto, Takashi. Cylindres dans les fibrations de Mori: formes du volume quintique de del Pezzo. Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2377-2393. doi : 10.5802/aif.3297. http://www.numdam.org/articles/10.5802/aif.3297/

[1] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI | MR | Zbl

[2] Dubouloz, Adrien; Kishimoto, Takashi Explicit biregular/birational geometry of affine threefolds: completions of 𝔸 3 into del Pezzo fibrations and Mori conic bundles, Algebraic varieties and automorphism groups (Advanced Studies in Pure Mathematics), Volume 75, Mathematical Society of Japan, 2017, pp. 49-71 | DOI | MR | Zbl

[3] Dubouloz, Adrien; Kishimoto, Takashi Cylinders in del Pezzo fibrations, Isr. J. Math., Volume 225 (2018) no. 2, pp. 797-815 | DOI | MR | Zbl

[4] Dubouloz, Adrien; Kishimoto, Takashi Deformations of 𝔸 1 -cylindrical varieties, Math. Ann., Volume 373 (2019) no. 3-4, pp. 1135-1149 | DOI | MR | Zbl

[5] Furushima, Mikio The complete classification of compactifications of C 3 which are projective manifolds with the second Betti number one, Math. Ann., Volume 297 (1993) no. 4, pp. 627-662 | DOI | MR | Zbl

[6] Furushima, Mikio; Nakayama, Noboru The family of lines on the Fano threefold V 5 , Nagoya Math. J., Volume 116 (1989), pp. 111-122 | DOI | Zbl

[7] Furushima, Mikio; Nakayama, Noboru A new construction of a compactification of C 3 , Tôhoku Math. J., Volume 41 (1989) no. 4, pp. 543-560 | DOI | MR | Zbl

[8] Iliev, Atanas The Fano surface of the Gushel’ threefold, Compos. Math., Volume 94 (1994) no. 1, pp. 81-107 | MR | Zbl

[9] Iliev, Atanas Lines on the Gushel’ threefold, Indag. Math., New Ser., Volume 5 (1994) no. 3, pp. 307-320 | DOI | MR | Zbl

[10] Iskovskih, Vasily A. Anticanonical models of three-dimensional algebraic varieties, Current problems in mathematics, Vol. 12 (Russian), VINITI, 1979, pp. 59-157

[11] Kishimoto, Takashi; Prokhorov, Yuri; Zaidenberg, Mikhail Group actions on affine cones, Affine algebraic geometry (CRM Proceedings & Lecture Notes), Volume 54, American Mathematical Society, 2011, pp. 123-163 | DOI | MR

[12] Kishimoto, Takashi; Prokhorov, Yuri; Zaidenberg, Mikhail 𝔾 a -actions on affine cones, Transform. Groups, Volume 18 (2013) no. 4, pp. 1137-1153 | DOI | MR | Zbl

[13] Kishimoto, Takashi; Prokhorov, Yuri; Zaidenberg, Mikhail Affine cones over Fano threefolds and additive group actions, Osaka J. Math., Volume 51 (2014) no. 4, pp. 1093-1112 | MR | Zbl

[14] Kolpakov-Miroshnichenko, I. Ya.; Prokhorov, Yuri Construction of the rationality of fields of invariants of some finite four-dimensional linear groups that are connected with Fano threefolds, Mat. Zametki, Volume 51 (1992) no. 1, pp. 114-117 | DOI | MR | Zbl

[15] Kuznetsov, Alexander G.; Prokhorov, Yuri; Shramov, Constantin A. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Jpn. J. Math., Volume 13 (2018) no. 1, pp. 109-185 | DOI | MR | Zbl

[16] Micali, Artibano Sur les algèbres universelles, Ann. Inst. Fourier, Volume 14 (1964) no. 2, pp. 33-87 | DOI | Zbl

[17] Mukai, Shigeru; Umemura, Hiroshi Minimal rational threefolds, Algebraic geometry (Tokyo/Kyoto, 1982) (Lecture Notes in Mathematics), Volume 1016, Springer, 1983, pp. 490-518 | DOI | MR | Zbl

[18] Prokhorov, Yuri Fano threefolds of genus 12 and compactifications of C 3 , Algebra Anal., Volume 3 (1991) no. 4, pp. 162-170 | MR | Zbl

[19] Prokhorov, Yuri; Zaidenberg, Mikhail Examples of cylindrical Fano fourfolds, Eur. J. Math., Volume 2 (2016) no. 1, pp. 262-282 | DOI | MR | Zbl

[20] Prokhorov, Yuri; Zaidenberg, Mikhail New examples of cylindrical Fano fourfolds, Algebraic varieties and automorphism groups (Advanced Studies in Pure Mathematics), Volume 75, Mathematical Society of Japan, 2017, pp. 443-463 | DOI | MR | Zbl

[21] Serre, Jean-Pierre Sur les modules projectifs, Séminaire Dubreil. Algèbre et théorie des nombres (1960/61), Secrétariat mathématique, 1963, pp. 1-16 | Zbl

[22] Takeuchi, Kiyohiko Some birational maps of Fano 3-folds, Compos. Math., Volume 71 (1989) no. 3, pp. 265-283 | MR | Zbl

Cited by Sources: