Moduli spaces and the inverse Galois problem for cubic surfaces

We study the moduli space $\widetilde{\text{calM}}$ of marked cubic surfaces. By classical work of A.,B. Coble, this has a compactification $\tilde{M}$, which is linearly acted upon by the group $W(E_6)$. $\tilde{M}$ is given as the intersection of 30 cubics in ${\text{bP}}^9$. For the morphism $\widetilde{calM} \to \bP(1,2,3,4,5)$ forgetting the marking, followed by Clebsch’s invariant map, we give explicit formulas. I.e., Clebsch’s invariants are expressed in terms of Coble’s irrational invariants. As an application, we give an affirmative answer to the inverse Galois problem for cubic surfaces over $\text{bbQ}$.