Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

G-torsors and universal torsors over nonsplit del Pezzo surfaces

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 Néron-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.

G-torsors and universal torsors over nonsplit del Pezzo surfaces

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 in Pic , or a form of it containing the Néron-Severi torus. Let be the G-torsor over obtained by extension of structure group from a universal torsor over . We prove that does not descend to S unless does. This is in contrast to a result of Friedman and Morgan that such always descend to singular del Pezzo surfaces over from their desingularizations.

Paper Structure

This paper contains 4 sections, 7 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Torsors
  3. Automorphisms of $T$-torsors and $G$-torsors
  4. Proof of the main results

Key Result

Theorem 1

Let $G$ be a reductive group over $K$ with maximal torus $\iota: T \hookrightarrow G$ such that Let $\mathcal{T}$ be a universal $T_L$-torsor over $S_L$, and let $\mathcal{G}:=(\iota_L)_*\mathcal{T}$ be the $G_L$-torsor over $S_L$ obtained by extension of structure group. Then the groupoid of $G$-torsors $\mathcal{G}^\circ$ over $S$ such that $\mathcal{G}^\circ_L \cong \mathcal{G}$ is equivale

Theorems & Definitions (19)

  • Theorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • ...and 9 more