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/3Z deck group action. We compute the image of the monodromy map rho induced by the action of pi(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 (E6). Our main result is that rho 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.