Equivariantly normal varieties for diagonalizable group actions

TL;DR

This paper develops criteria and structural descriptions for $G$-normal varieties under diagonalizable group actions, extending classical concepts and providing a Hurwitz formula analogue for linearly reductive groups.

Contribution

It introduces a criterion for $G$-normality, describes local structures in codimension 1, and extends the Hurwitz formula to $G$-normal varieties with linearly reductive groups.

Findings

Criterion for $G$-normality established

Local structure of $G$-normal varieties described

Hurwitz formula extended to $G$-normal varieties

Abstract

Given a finite group scheme $G$ over a field and a $G$-variety $X$, we obtain a criterion for $X$ to be $G$-normal in the sense of \cite{Br24}. When $G$ is diagonalizable, we describe the local structure of $G$-normal varieties in codimension $1$ and their dualizing sheaf. As an application, we obtain a version of the Hurwitz formula for $G$-normal varieties, where $G$ is linearly reductive.