Monodromy of the family of cubic surfaces branching over smooth cubic curves (Q2675316)

The author studies the family of smooth cubic surfaces which can be realized as threefold-branched covers of \(\mathbb{P}^2\), with branch locus equal to a smooth cubic curve. This family is parametrized by the space \(\mathcal{U}_3\) of smooth cubic curves in \(\mathbb{P}^2\) and each surface is equipped with a \(\mathbb{Z}/3\mathbb{Z}\) deck group action. The paper continues results of Klein and Jordan about the fiber bundle defined by the universal cubic surface. The latter authors have shown that the Galois group of the equation for the 27 lines on a cubic is precisely \(W(E_6)\), the Weyl group of \(E_6\), which coincides with the image of the monodromy homomorphism induced by the universal family \(\mathcal{E}_{3,3}\rightarrow \mathcal{U}_{3,3}\). The author computes the image of the monodromy map \(\rho\) induced by the action of \(\pi _1(\mathcal{U}_3)\) on the 27 lines contained on the cubic surfaces of this family. One knows that this image is contained in the Weyl group \(W(E_6)\). The author shows that \(\rho\) is surjective onto the centralizer of the image of a generator of the deck group. The 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.