Galois points for a normal hypersurface

TL;DR

This paper investigates Galois points on hypersurfaces, establishing bounds on their number, characterizing when the hypersurface is a cone, and applying hyperplane section theorems across different characteristics.

Contribution

It provides a sharp upper bound for the number of Galois points, characterizes hypersurfaces with infinite Galois points as cones, and proves a hyperplane section theorem applicable in all characteristics.

Findings

Bound on the number of Galois points in terms of dimensions

Hypersurfaces with infinite Galois points are cones

Classification of Galois groups for Galois points

Abstract

We study Galois points for a hypersurface $X$ with $\dim {\rm Sing}(X) \le \dim X-2$. The purpose of this article is to determine the set $\Delta(X)$ of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of $\dim X$ and $\dim {\rm Sing}(X)$ if $\Delta(X)$ is a finite set, and prove that $X$ is a cone if $\Delta(X)$ is infinite. To achieve our purpose, we need a certain hyperplane section theorem on Galois point. We prove this theorem in arbitrary characteristic. On the other hand, the hyperplane section theorem has other important applications: For example, we can classify the Galois group induced from a Galois point in arbitrary characteristic and determine the distribution of Galois points for a Fermat hypersurface of degree $p^e+1$ in characteristic $p>0$.