On plane quartics with a Galois invariant Steiner hexad

TL;DR

This paper constructs plane quartics with specific Galois actions on their bitangents, focusing on Galois invariant Steiner hexads, and applies this to solve a case of the inverse Galois problem for degree two del Pezzo surfaces.

Contribution

It introduces a method to construct plane quartics with a prescribed Galois action on bitangents, specifically for Galois invariant Steiner hexads, and applies it to inverse Galois problems.

Findings

Constructed plane quartics with Galois invariant Steiner hexads.

Solved the inverse Galois problem for certain degree two del Pezzo surfaces.

Established a link between Galois actions and geometric configurations.

Abstract

We describe a construction of plane quartics with prescribed Galois operation on the 28 bitangents, in the particular case of a Galois invariant Steiner hexad. As an application, we solve the inverse Galois problem for degree two del Pezzo surfaces in the corresponding particular case.