Linear automorphisms of smooth hypersurfaces giving Galois points

TL;DR

This paper characterizes the existence of Galois points on smooth hypersurfaces of degree at least four using linear automorphisms, providing necessary and sufficient conditions for such points.

Contribution

It introduces a criterion based on linear automorphisms to determine Galois points on smooth hypersurfaces, advancing understanding of their geometric and algebraic properties.

Findings

Provides necessary and sufficient conditions for Galois points

Uses linear automorphisms to characterize Galois points

Enhances understanding of hypersurface automorphisms

Abstract

Let $X$ be a smooth hypersurface $X$ of degree $d\geq4$ in a projective space $\mathbb P^{n+1}$. We consider a projection of $X$ from $p\in\mathbb P^{n+1}$ to a plane $H\cong\mathbb P^n$. This projection induces an extension of function fields $\mathbb C(X)/\mathbb C(\mathbb P^n)$. The point $p$ is called a Galois point if the extension is Galois. In this paper, we will give a necessary and sufficient conditions for $X$ to have Galois points by using linear automorphisms.