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.

Details

Title

Monodromy of Some Families of Cubic and K3 Surfaces

Author

Medrano Martin del Campo, Adan : University of Chicago