2009/12/03 by Mathieu Huruguen, Huruguen, Mathieu
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR) #Polynomial and algebraic computation #math.AG #math.GR
paper · pdf · doi:10.48550/arxiv.0912.0676
openalex publication_date 2009/12/03 · arxiv created 2011/04/14 · arxiv updated 2011/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 X0 over \mr of rank 2 under the action of SU(2,1) and a smooth embedding of X0 whose fan is Galois-stable but which admits no \mr-form.