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)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik
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

Elektronisch veröffentlicht (E-Pub)

Electronic version(s)

https://arxiv.org/abs/2109.08137 (Access: Offen)