On the inverse Galois problem for del Pezzo surfaces of degree 1

TL;DR

This paper addresses the inverse Galois problem for del Pezzo surfaces of degree 1 over finite fields, solving it for most types and identifying minimal fields of existence.

Contribution

It provides a complete solution for 85 of 112 types and determines the smallest fields of existence for all types, advancing understanding of these surfaces.

Findings

Solved the inverse Galois problem for 85 of 112 types over finite fields.

Determined the minimal field of existence for all 112 types.

Presented an example of a del Pezzo surface with multiple generalized Eckardt points in characteristic 2.

Abstract

We solve the inverse Galois problem for del Pezzo surfaces of degree 1 over finite fields completely for 85 of the 112 possible types. We also determine for all 112 types the smallest field of existence. As an aside, we provide an example of a del Pezzo surface of degree 1 in characteristic 2 with more than one generalized Eckardt point.