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Adjoint representation of $E_{8}$ and del Pezzo surfaces of degree $1$

Vera V. Serganova; Alexei N. Skorobogatov — 2011

Annales de l’institut Fourier

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...

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Adjoint representation of E 8 and del Pezzo surfaces of degree 1

Adjoint representation of $E_{8}$ and del Pezzo surfaces of degree $1$