Sixteen points in $\mathbb{P}^4$ and the inverse Galois problem for del Pezzo surfaces of degree one

Abstract

A del Pezzo surface of degree one defined over the rationals has 240 exceptional curves. These curves are permuted by the action of the absolute Galois group. We show how a solution to the classical inverse Galois problem for a subgroup of the Weyl group of type $D_8$ gives rise to a solution of the inverse Galois problem for the action of this subgroup on the 240 exceptional curves. A del Pezzo surface of degree one with such a Galois action contains a Galois invariant sublattice of type $D_8$ within its Picard lattice; this can be characterized in terms of a certain set of sixteen points in $\mathbb{P}^4$.