Monodromy of the families of del Pezzo and $K3$ surfaces branching over smooth quartic curves

TL;DR

This paper computes the monodromy groups of two families of surfaces derived from cyclic branched covers over smooth quartic curves, revealing connections to Weyl groups and arithmetic lattices.

Contribution

It explicitly determines the monodromy groups for families of degree 2 del Pezzo and K3 surfaces, linking geometric structures to algebraic groups.

Findings

Monodromy group for del Pezzo surfaces is the Weyl group W(E7).

Monodromy group for K3 surfaces is an arithmetic lattice U(h_{L_{-}}).

Results connect surface automorphisms with algebraic group theory.

Abstract

Two families of surfaces 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.