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

TL;DR

This paper links the inverse Galois problem for certain Weyl groups to the Galois action on the exceptional curves of degree one del Pezzo surfaces, using configurations of sixteen points in projective 4-space.

Contribution

It establishes a method to realize Galois groups as automorphisms of del Pezzo surfaces via configurations of sixteen points in projective space.

Findings

Galois group actions correspond to configurations of points in projective space.

A Galois invariant sublattice of type D8 is characterized by these point configurations.

The approach connects classical inverse Galois problems with algebraic geometry of del Pezzo surfaces.

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 $\mathbb{P}^4$.