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Abstract
This thesis is a compilation of three papers.
In Chapter 1, we introduce the general setup for these papers, which concern families of branched covers of $\mathbb{P}^{2}$ branching over smooth curves of a fixed degree.
In Chapter 2, we 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.
In Chapter 3, we study the two families of surfaces which arise from considering cyclic branched covers of $\mathbb{P}^{2}$ over smooth quartic curves. These consist of degree 2 del Pezzo surfaces with a $\mathbb{Z}/2\mathbb{Z}$ action and $K3$ surfaces with a $\mathbb{Z}/4\mathbb{Z}$ action. We compute the monodromy groups of both families. In the first case, we obtain the Weyl group $W\left(E_{7}\right)$, corresponding to the automorphisms of the $56$ lines contained in a degree $2$ del Pezzo surface. In the second case we obtain an arithmetic lattice: the unitary group $U\left(h_{L_{-}}\right)$ of a type $\left(1, 6\right)$ quadratic form over $\mathbb{Z}\left[i\right]$ by building on results of Kondo and Allcock, Carlson, Toledo.
In Chapter 4, we study families of surfaces which arise from cyclic branched covers of $\mathbb{P}^{2}$ over smooth sextic curves. These consist of surfaces with a $\mathbb{Z}/d\mathbb{Z}$ action for $d=2, 3, 6$. We bound the monodromy groups of the families corresponding to $d=2,3$. In doing so, we conjecture equivalent characterizations of the moduli of smooth sextic curves as quotients by building on results of Looijenga and Allcock, Carlson, Toledo.