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G-torsors and universal torsors over nonsplit del Pezzo surfaces

authored by
Ulrich Derenthal, Norbert Hoffmann
Abstract

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.

Organisation(s)
Institute of Algebra, Number Theory and Discrete Mathematics
Riemann Center for Geometry and Physics
External Organisation(s)
Mary Immaculate College
Type
Preprint
No. of pages
9
Publication date
16.09.2021
Publication status
E-pub ahead of print
Electronic version(s)
https://arxiv.org/abs/2109.08137 (Access: Open)