Monodromy of the family of cubic surfaces branching over smooth cubic curves

TL;DR

This paper investigates the monodromy action on the 27 lines of cubic surfaces obtained as threefold covers of the plane branched over smooth cubic curves, showing the monodromy map is surjective onto a specific subgroup of the Weyl group E6.

Contribution

It explicitly computes the monodromy image for this family of cubic surfaces and proves its surjectivity onto the centralizer subgroup, revealing new insights into the symmetry structure of these surfaces.

Findings

Monodromy map is surjective onto the centralizer of a deck group generator.

The monodromy group is contained in the Weyl group E6.

The proof uses relations between inflection points and lines on cubic surfaces.

Abstract

Consider 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. We compute the image of the monodromy map $\rho$ induced by the action of $\pi_{1}\left(\mathcal{U}_{3}\right)$ 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 $W\left(E_{6}\right)$. 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.