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

Cao, Yang

Épijournal de Géométrie Algébrique, Tome 2 (2018)

Résumé

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 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.

Détail

DOI : 10.46298/epiga.2018.volume2.3302