Galois points for a normal hypersurface

Abstract:

A Galois point for a hypersurface is a point from which the projection induces a Galois extension of function fields. The purpose of this article is to determine the set $\Delta (X)$ of Galois points for a hypersurface $X$ with $\dim \textrm {Sing}(X) \le \dim X-2$ in characteristic zero: In fact, if $X$ is not a cone, we give a sharp upper bound for the cardinality of $\Delta (X)$ in terms of $\dim X$ and $\dim \textrm {Sing}(X)$, and completely classify $X$ attaining the bound. We determine $\Delta (X)$ also when $X$ is a cone. To achieve our purpose, we prove a certain hyperplane section theorem on a Galois point in arbitrary characteristic. The hyperplane section theorem has other important applications: For example, we can classify the Galois group induced by a Galois point in arbitrary characteristic and determine the distribution of Galois points for a Fermat hypersurface of degree $p^e+1$ in characteristic $p>0$.