Monodromy of the family of Cubic Surfaces branching over Smooth Cubic Curves

Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers of ℙ 2 , with branch locus equal to a smooth cubic curve. This family is parametrized by the space 𝒰 3 of smooth cubic curves in ℙ 2 and each surface is equipped with a ℤ / 3 ℤ deck group action. We compute the image of the monodromy map ρ induced by the action of π 1 ( 𝒰 3 ) on the 27 lines contained on the cubic surfaces of this family. Due to a classical result, this image is contained in the Weyl group W E 6 . Our main result is that ρ is surjective onto the centralizer of the image a of a generator of the deck group. Our proof is mainly computational, and relies on the relation between the 9 inflection points in a cubic curve and the 27 lines contained in the cubic surface branching over it.