Papers › Monodromy of the family of cubic surfaces branching over smooth cubic curves
Monodromy of the family of cubic surfaces branching over smooth cubic curves
Adán Medrano Martín del Campo
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Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers of ℙ², with branch locus equal to a smooth cubic curve. This family is parametrized by the space 𝒰₃ of smooth cubic curves in ℙ² and each surface is equipped with a ℤ/3ℤ deck group action. We compute the image of the monodromy map ρ induced by the action of π₁(𝒰₃) 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 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.
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