Adjoint representation of $\text{\upshape E}_8$ and del Pezzo surfaces of degree $1$

Serganova, Vera V.; Skorobogatov, Alexei N.

doi:10.5802/aif.2676

Adjoint representation of

E_{8}

and del Pezzo surfaces of degree

1

[La représentation adjointe de

E_{8}

et les surfaces de del Pezzo de degré

1

]

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)

Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2337-2360

Abstract (VO)
Résumé

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 $X$ .

EuDML MR Zbl

DOI : 10.5802/aif.2676

Classification : 14J26, 14M17, 22E46
Keywords: Universal torsors, del Pezzo surfaces, Lie groups, homogeneous spaces
Mots-clés : torseurs universels, surfaces de del Pezzo, groupes de Lie, espaces homogènes

Affiliations des auteurs :

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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     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},
     pages = {2337--2360},
     year = {2011},
     publisher = {Association des Annales de l'Institut Fourier},
     volume = {61},
     number = {6},
     doi = {10.5802/aif.2676},
     zbl = {1257.14025},
     mrnumber = {2976314},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2676/}
}

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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, Tome 61 (2011) no. 6, pp. 2337-2360. doi: 10.5802/aif.2676

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