Galois Groups of Symmetric Cubic Surfaces

The Galois group of a family of cubic surfaces is the monodromy group of the 27 lines of its generic fibre. We describe a method to compute this group for linear systems of cubic surfaces using certified numerical computations. Applying this to all families which are invariant under the action of a subgroup of $S_5$, we find that the Galois group is often much smaller than the Weyl group $W(E_6)$. As a byproduct, we compute the discriminants of these~families. Our method allows to compute the monodromy representation on homology of any family of generically smooth projective hypersurfaces. To illustrate this broader scope, we include computations for symmetric quartic surfaces.