Adjoint representation of E 8 and del Pezzo surfaces of degree 1
Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2337-2360.

Let X be a del Pezzo surface of degree 1, and let G be the simple Lie group of type E 8 . We construct a locally closed embedding of a universal torsor over X into the G-orbit of the highest weight vector of the adjoint representation. This embedding is equivariant with respect to the action of the Néron-Severi torus T of X identified with a maximal torus of G extended by the group of scalars. Moreover, the T-invariant hyperplane sections of the torsor defined by the roots of G are the inverse images of the 240 exceptional curves on X.

Soit X une surface de del Pezzo de degré 1, et soit G un groupe de Lie simple de type E 8 . Nous montrons que tout torseur universel sur X est un sous-ensemble localement fermé de la G-orbite d’un vecteur du plus grand point de la représentation adjointe. Ce plongement est équivariant par rapport à l’action du tore de Néron–Severi T de X, identifié avec un tore maximal de l’extension de G par le groupe de scalaires. En outre, les sections hyperplanes T-invariantes du torseur définies par les racines de G sont les images réciproques des 240 courbes exceptionnelles de X.

DOI: 10.5802/aif.2676
Classification: 14J26,  14M17,  22E46
Keywords: Universal torsors, del Pezzo surfaces, Lie groups, homogeneous spaces
Serganova, Vera V. 1; Skorobogatov, Alexei N. 2

1 University of California Department of Mathematics Berkeley, CA, 94720-3840 (USA)
2 Imperial College London Department of Mathematics South Kensington Campus SW7 2BZ England, (U.K.) Institute for the Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetnyi Moscow, 127994 (Russia)
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Serganova, Vera V.; Skorobogatov, Alexei N. Adjoint representation of $\text{\upshape E}_8$ and del Pezzo surfaces of degree $1$. Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2337-2360. doi : 10.5802/aif.2676. http://archive.numdam.org/articles/10.5802/aif.2676/

[1] Bourbaki, N. Groupes et algèbres de Lie, Chapitres IV-VIII, Masson, Paris, 1975, 1981 | MR | Zbl

[2] Fulton, W.; Harris, J. Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer-Verlag, 1991 | MR | Zbl

[3] Hartshorne, R. Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, 1977 | MR | Zbl

[4] Lurie, J. On simply laced Lie algebras and their minuscule representations, Comm. Math. Helv., Volume 76 (2001), pp. 515-575 | DOI | MR | Zbl

[5] Manin, Yu.I. Cubic forms, North-Holland, 1986 | MR | Zbl

[6] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Springer-Verlag, 1994 | MR | Zbl

[7] Serganova, V.V.; Skorobogatov, A.N. Del Pezzo surfaces and representation theory, Algebra Number Theory, Volume 1 (2007), pp. 393-419 | DOI | MR | Zbl

[8] Serganova, V.V.; Skorobogatov, A.N. On the equations for universal torsors over del Pezzo surfaces, J. Inst. Math. Jussieu, Volume 9 (2010), pp. 203-223 | DOI | MR | Zbl

[9] Sturmfels, B.; Xu, Z. Sagbi Bases of Cox–Nagata Rings, J. Eur. Math. Soc., Volume 12 (2010), pp. 429-459 | DOI | MR | Zbl

[10] Testa, D.; Várilly-Alvarado, A.; Velasco, M. Cox rings of degree one del Pezzo surfaces, Algebra Number Theory, Volume 3 (2009), pp. 729-761 | DOI | MR | Zbl

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