Actions of finite group schemes on curves

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 Gnormalization. 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.