Adjoint representation of E 8 and del Pezzo surfaces of degree 1
Annales de l'Institut Fourier, Volume 61 (2011) no. 6, p. 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 : https://doi.org/10.5802/aif.2676
Classification:  14J26,  14M17,  22E46
Keywords: Universal torsors, del Pezzo surfaces, Lie groups, homogeneous spaces
@article{AIF_2011__61_6_2337_0,
     author = {Serganova, Vera V. and Skorobogatov, Alexei N.},
     title = {Adjoint representation of $\text{\upshape E}\_8$ and del Pezzo surfaces of degree $1$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     pages = {2337-2360},
     doi = {10.5802/aif.2676},
     mrnumber = {2976314},
     zbl = {1257.14025},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_6_2337_0}
}
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://www.numdam.org/item/AIF_2011__61_6_2337_0/

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