On plane quartics with a Galois invariant Cayley octad

TL;DR

This paper constructs plane quartics with specific Galois actions on their bitangents, focusing on Galois invariant Cayley octads, and applies this to solve the inverse Galois problem for certain degree two del Pezzo surfaces.

Contribution

It introduces a method to construct plane quartics with a prescribed Galois action on bitangents in the case of Galois invariant Cayley octads, advancing understanding of Galois representations.

Findings

Constructed plane quartics with prescribed Galois action on bitangents.

Solved the inverse Galois problem for degree two del Pezzo surfaces in a specific case.

Provided explicit examples of Galois invariant Cayley octads.

Abstract

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