Automorphisms and forms of toric quotients of homogeneous spaces

A. N. Skorobogatov

Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1521-1536

Résumé

We compute the automorphism group of the quotient of a generalized Grassmannian $G/P$ by the action of a maximal torus of the semisimple group $G$. We classify the twisted forms of such quotients, that is, varieties isomorphic to these quotients over an algebraic closure of the base field. It is proved that all such forms are unirational. Bibliography: 20 titles.

Détail

Keywords: homogeneous spaces, twisted forms, torsors.
Mots-clés : tori

Bibliographie

[1] J-L. Colliot-Thélène, J-J. Sansuc, “La descente sur les variétés rationnelles. II”, Duke Math. J., 54:2 (1987), 375–492 | DOI | MR | Zbl

[2] A. N. Skorobogatov, Torsors and rational points, Cambridge Tracts in Math., 144, Cambridge Univ. Press, Cambridge, 2001 | MR | Zbl

[3] I. Dolgachev, D. Ortland, Point sets in projective spaces and theta functions, Astérisque, 165, Société Mathématique de France, Paris, 1988 | MR | Zbl

[4] A. N. Skorobogatov, “On a theorem of Enriques–Swinnerton–Dyer”, Ann. Fac. Sci. Toulouse Math. (6), 2:3 (1993), 429–440 | MR | Zbl

[5] M. M. Kapranov, “Chow quotients of Grassmannians. I”, I. M. Gel'fand seminar. Part 2 (Moscow, 1993), Adv. Soviet Math., 16, Amer. Math. Soc., Providence, RI, 1993, 29–110 | MR | Zbl

[6] V. V. Serganova, A. N. Skorobogatov, “Del pezzo surfaces and representation theory”, Algebra Number Theory, 1:4 (2007), 393–419 | MR | Zbl

[7] V. V. Serganova, A. N. Skorobogatov, “On the equations for universal torsors over del Pezzo surfaces”, J. Inst. Math. Jussieu (to appear) | DOI

[8] M. Demazure, “Automorphismes et déformations des variétés de Borel”, Invent. Math., 39:2 (1977), 179–186 | DOI | MR | Zbl

[9] Ph. Gille, “Type des tores maximaux des groupes semi-simples”, J. Ramanujan Math. Soc., 19:3 (2004), 213–230 | MR

[10] M. S. Raghunathan, “Tori in quasi-split groups”, J. Ramanujan Math. Soc., 19:4 (2004), 281–287 | MR | Zbl

[11] D. Harari, A. Skorobogatov, “Non-abelian descent and the arithmetic of Enriques surfaces”, Int. Math. Res. Not., 52 (2005), 3203–3228 | DOI | MR | Zbl

[12] J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Math., 5, Springer-Verlag, Berlin–Heidelberg–New York, 1994 | DOI | MR | MR | Zbl | Zbl

[13] J. Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | MR | Zbl

[14] Zh. Dedonne, Dzh. Kerrol, D. Mamford, Geometricheskaya teoriya invariantov, Mir, M., 1974 ; J. A. Dieudonné, J. B. Carrell, Invariant theory, old and new, Academic Press, New York–London, 1971 ; D. Mumford, Geometric invariant theory, Springer-Verlag, Berlin–New York, 1965 | MR | Zbl | MR | Zbl | MR | Zbl

[15] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York–Heidelberg, 1977 | MR | MR | Zbl | Zbl

[16] I. Dolgachev, Lectures on invariant theory, London Math. Soc. Lecture Note Ser., 296, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl

[17] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968 | MR | MR | Zbl | Zbl

[18] N. Bourbaki, Éléments de mathématique., Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII–VIII, Hermann, Paris, 1975 | MR | MR | Zbl

[19] S. Senthamarai Kannan, “Torus quotients of homogeneous spaces – II”, Proc. Indian Acad. Sci. Math. Sci., 109:1 (1999), 23–39 | DOI | MR | Zbl

[20] V. Platonov, A. Rapinchuk, Algebraic groups and number theory, Pure Appl. Math., 139, Academic Press, Boston, MA, 1994 | MR | MR | Zbl | Zbl