Galois group of exceptional curves on the generic del Pezzo surface

We prove that the Galois action on the exceptional curves on the generic del Pezzo surface of degree $d$ is maximal for all degrees $d$ and over any field $k$. As a consequence of the case $d=3$, we deduce that over ${\mathbb{F}}_q(u)$, 100% of cubic surfaces have no Brauer-Manin obstruction.