Quartic surfaces with a Galois point and Eisenstein K3 surfaces

TL;DR

This paper establishes a correspondence between certain quartic surfaces with Galois points and Eisenstein K3 surfaces, also characterizing those with the maximum number of Galois points as singular K3 surfaces.

Contribution

It introduces a one-to-one correspondence between smooth quartic surfaces with an inner Galois point and Eisenstein K3 surfaces of type (4,3), and characterizes quartic surfaces with maximum Galois points.

Findings

Established a correspondence between quartic surfaces and Eisenstein K3 surfaces.

Characterized quartic surfaces with 8 Galois points as singular K3 surfaces.

Provided a classification related to Galois points on quartic surfaces.

Abstract

We prove that there exists a one to one correspondence between smooth quartic surfaces with an inner Galois point and Eisenstein $K3$ surfaces of type $(4, 3)$. Furthermore we characterize the quartic surface with 8 (the maximum number) inner Galois points as a singular $K3$ surface.