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

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 $H_{nr}^3({\mathcal{T}}^c,\mathbb{Q}\mathrm{/}\mathbb{Z}(2))\mathrm{/}{H}^3(k,\mathbb{Q}\mathrm{/}\mathbb{Z}(2))$ 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.