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G-torsors and universal torsors over nonsplit del Pezzo surfaces. (arXiv:2109.08137v2 [math.AG] UPDATED)

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Let S be a smooth del Pezzo surface that is defined over a field K and splits
over a Galois extension L. Let G be either the split reductive group given by
the root system of $S_L$ in Pic $S_L$, or a form of it containing the
N\'eron-Severi torus. Let $\mathcal{G}$ be the G-torsor over $S_L$ obtained by
extension of structure group from a universal torsor $\mathcal{T}$ over $S_L$.
We prove that $\mathcal{G}$ does not descend to S unless $\mathcal{T}$ does.
This is in contrast to a result of Friedman and Morgan that such $\mathcal{G}$
always descend to singular del Pezzo surfaces over $\mathbb{C}$ from their
desingularizations.

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