Troisi\`eme groupe de cohomologie non ramifi\'ee des torseurs universels sur les surfaces rationnelles

TL;DR

This paper investigates the third unramified cohomology groups of universal torsors over rational surfaces, providing conditions under which these groups vanish or are reduced, thus contributing to understanding their birational properties.

Contribution

It offers new criteria based on Galois structures for the vanishing or reduction of third unramified cohomology groups of universal torsors over rational surfaces.

Findings

Vanishing of certain unramified cohomology groups for generalised Ch\^atelet surfaces.

Reduction to 2-primary part of the cohomology group for del Pezzo surfaces of degree ≥ 2.

Provides conditions linking Galois structures to cohomological properties.

Abstract

Let $k$ a field of characteristic zero. Let $X$ be a smooth, projective, geometrically rational $k$-surface. Let $\mathcal{T}$ be a universal torsor over $X$ with a $k$-point et $\mathcal{T}^c$ a smooth compactification of $\mathcal{T}$. There is an open question: is $\mathcal{T}^c$ $k$-birationally equivalent to a projective space? We know that the unramified cohomology groups of degree 1 and 2 of $\mathcal{T}$ and $\mathcal{T}^c$ are reduced to their constant part. For the analogue of the third cohomology groups, we give a sufficient condition using the Galois structure of the geometrical Picard group of $X$. This enables us to show that $H^{3}_{nr}(\mathcal{T}^{c},\mathbb{Q}/\mathbb{Z}(2))/H^3(k,\mathbb{Q}/\mathbb{Z}(2))$ vanishes if $X$ is a generalised Ch\^atelet surface and that this group is reduced to its $2$-primary part if $X$ is a del Pezzo surface of degree at least 2.