Troisième groupe de cohomologie non ramifiée des torseurs universels sur les surfaces rationnelles

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 $H_{nr}^3({\mathcal{T}}^c,\mathbb{Q}/\mathbb{Z}(2))/H^3(k,\mathbb{Q}/\mathbb{Z}(2))$ vanishes if $X$ is a generalised Châtelet surface and that this group is reduced to its $2$-primary part if $X$ is a del Pezzo surface of degree at least 2.

27 page, in French

References in corpus (1)

• Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Cited by in corpus (1)

• Noether's problem and descent