Actions of finite group schemes on curves

TL;DR

The paper introduces the concept of G-normalization for finite group scheme actions on curves, establishing the existence and uniqueness of G-normal models and their relevance to surface questions in positive characteristic.

Contribution

It defines G-normalization for G-actions on curves and proves the existence and uniqueness of G-normal models, linking to surface theory in positive characteristic.

Findings

Every G-action on a curve has a unique G-normal model.

G-normal curves are characterized by invertible ideal sheaves of orbits.

G-normal models are essential in questions about surfaces in positive characteristic.

Abstract

Every action of a finite group scheme $G$ on a variety admits a projective equivariant model, but not necessarily a normal one. As a remedy, we introduce and explore the notion of $G$-normalization. In particular, every curve equipped with a $G$-action has a unique projective $G$-normal model, characterized by the invertibility of ideal sheaves of all orbits. Also, $G$-normal curves occur naturally in some questions on surfaces in positive characteristics.