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Let $X$ be a del Pezzo surface of degree $1$, and let $G$ be the simple Lie group of type ${\text{E}}_8$. 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...

Let $X$ be an algebraic variety defined over a field $k$ of characteristic $0$, and let $Y$ be an $X$-torsor under a torus. We compute the Brauer group of $Y$. In the case of a number field $k$ we deduce results concerning the arithmetic of $X$.

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$, and let $\overline{X}$ and $\overline{Y}$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br $\left(\overline{X}\right)\oplus$ Br($\overline{Y})$ has finite index in the Galois invariant subgroup of Br$(\overline{X}\times \overline{Y})$. This implies that the cokernel of the natural map Br$\left(X\right)\oplus$ Br$\left(Y\right)\to$ Br$(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the product of...