Spherical Spaces

TL;DR

This paper introduces the concept of spherical spaces over arbitrary base schemes, explores their behavior in families, and classifies spherical embeddings over various fields, extending and simplifying existing results.

Contribution

It generalizes the notion of spherical varieties to arbitrary base schemes and provides new classification results for spherical embeddings over different fields.

Findings

Sphericity of subgroup schemes is an open and closed condition over arbitrary base schemes.

Spherical embeddings are classified over arbitrary fields.

The results extend and simplify previous classifications by Huruguen, Knop, and Roehrle.

Abstract

The notion of a spherical space over an arbitrary base scheme is introduced as a generalization of a spherical variety over an algebraically closed field. It is studied how the sphericity condition behaves in families. In particular it is shown that sphericity of subgroup schemes is an open and closed condition over arbitrary base schemes generalizing a result by Knop and Roehrle. Moreover spherical embeddings are classified over arbitrary fields generalizing and simplifying results by Huruguen.