Toric varieties and spherical embeddings over an arbitrary field

TL;DR

This paper explores the classification of toric varieties and spherical embeddings over arbitrary fields using combinatorial objects, providing new examples and conditions related to Galois-stability and field definitions.

Contribution

It characterizes Galois-stable fans and colored fans for varieties over arbitrary fields and constructs explicit examples illustrating limitations of field forms.

Findings

Galois-stable fans classify toric varieties over fields

Galois-stable colored fans classify spherical embeddings over fields

Examples of Galois-stable fans without corresponding field forms

Abstract

We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field $k$. Then we provide some situations where toric varieties over $k$ are classified by Galois-stable fans, and spherical embeddings over $k$ by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over $k$ whose fan is Galois-stable but which admits no $k$-form. In the spherical setting, we offer an example of a spherical homogeneous space $X_0$ over $\mr$ of rank 2 under the action of SU(2,1) and a smooth embedding of $X_0$ whose fan is Galois-stable but which admits no $\mr$-form.