G-torsors and universal torsors over nonsplit del Pezzo surfaces

TL;DR

This paper investigates the descent properties of G-torsors over nonsplit del Pezzo surfaces, revealing that certain torsors do not descend unless their universal torsors do, contrasting previous results over complex surfaces.

Contribution

It establishes new non-descent results for G-torsors over nonsplit del Pezzo surfaces, extending understanding of torsor behavior in algebraic geometry.

Findings

G-torsors over nonsplit del Pezzo surfaces generally do not descend to the surface itself.

Contrasts with prior results over complex surfaces where descent always occurs.

Provides conditions under which descent of torsors fails.

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.