Geodesic
Birational invariants for the torus without affect in a group of $F_4$ type
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 6 (2009), pp. 57-68 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the paper all the cohomological birational invariants for the torus without affect in a semisimple exceptional group of $F_4$ type are calculated. Kunyavski B. and Cortella A. have proved that this torus is not rational. We prove that the Picard group of a projective model for the studied torus is not cohomologically trivial. We find all the subgroups in Weyl group $W(F_4)$ for which the corresponding cohomological invariant is not trivial.
Mots-clés : algebraic torus, birational invariant
Keywords: cohomology, flasque resolution, semisimple group. 
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     title = {Birational invariants for the torus without affect in a~group of $F_4$ type},
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Yu. Yu. Krutikov. Birational invariants for the torus without affect in a group of $F_4$ type. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 6 (2009), pp. 57-68. http://geodesic.mathdoc.fr/item/VSGU_2009_6_a5/

[1] Voskresenskii V. E., “Maksimalnye tory bez affekta v poluprostykh algebraicheskikh gruppakh”, Matem. zametki, 44:3 (1988), 309–318 | MR

[2] Voskresenskii V. E., Kunyavskii B. E., O maksimalnykh torakh v poluprostykh algebraicheskikh gruppakh, Dep. v VINITI 5.03.84., No 1269, Kuibyshev, 1984

[3] Klyachko A. A., Pryamye slagaemye perestanovochnykh modulei i biratsionalnaya geometriya. Arifmetika i geometriya mnogoobrazii, Samara, 1992

[4] Cortella A., Kunyavski B., “Rationality problem for generic tori in simple groups”, J. Algebra, 225 (2000), 771–793 | DOI | MR | Zbl

[5] Popov S. Yu., “Reshetki Galua i ikh biratsionalnye invarianty”, Vestnik SamGU, 1998, no. 4(10), 71–83 | Zbl

[6] Lemire N., Popov V. L., Reichstein Z., “Cayley groups”, J. Amer. Math. Soc., 19 (2006), 921–967 | DOI | MR | Zbl

[7] Belova L. A., “Moduli chetvernoi gruppy Kleina i ikh kogomologicheskie invarianty”, Vestnik SamGU, 2008, no. 6(65), 59–70 | MR

[8] Voskresenskii V. E., Algebraicheskie tory, Nauka, M., 1977 | MR

[9] Voskresenskii V. E., “Proektivnye invariantnye modeli Demazyura”, Izvestiya AN SSSR. Ser.: Matematicheskaya, 46:2 (1982), 195–210 | MR

[10] Burbaki N., Gruppy i algebry Li. Gruppy Kokstera i sistemy Titsa. Sistemy kornei, Mir, M., 1972 | MR

[11] Dzh. Kassels, A. Frelikh(red.), Algebraicheskaya teoriya chisel, Mir, M., 1969 | MR

[12] Cohen H., A course in computational algebraic number theory, Springer, Berlin–Heidelberg–New York, 1996 | MR