Moduli spaces and the inverse Galois problem for cubic surfaces

TL;DR

This paper explores the structure of the moduli space of marked cubic surfaces, providing explicit formulas for invariants and solving the inverse Galois problem for these surfaces over the rationals.

Contribution

It explicitly relates Clebsch's invariants to Coble's invariants and proves the inverse Galois problem for cubic surfaces over ield.

Findings

Explicit formulas for Clebsch's invariants in terms of Coble's invariants.

Confirmation of the inverse Galois problem for cubic surfaces over ield.

Description of the moduli space as an intersection of 30 cubics in ield.

Abstract

We study the moduli space $\widetilde{\calM}$ of marked cubic surfaces. By classical work of A.\,B. Coble, this has a compactification $\widetilde{M}$, which is linearly acted upon by the group $W(E_6)$. $\widetilde{M}$ is given as the intersection of 30 cubics in $\bP^9$. 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 $\bbQ$.