TY - UNPB

T1 - G-torsors and universal torsors over nonsplit del Pezzo surfaces

AU - Derenthal, Ulrich

AU - Hoffmann, Norbert

PY - 2021/9/16

Y1 - 2021/9/16

N2 - 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.

AB - 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.

M3 - Preprint

BT - G-torsors and universal torsors over nonsplit del Pezzo surfaces

ER -