Toricness and smoothness criteria for spherical varieties

TL;DR

This paper establishes numerical criteria to determine when a spherical variety is toric or smooth along G-orbits, using spherical skeletons, thereby simplifying and improving classical smoothness tests.

Contribution

It introduces new numerical conditions based on spherical skeletons for toricness and smoothness of spherical varieties, removing reliance on external reference tables.

Findings

Provides equivalent numerical conditions for toric structures.

Offers an improved smoothness criterion independent of external tables.

Simplifies the analysis of spherical varieties using spherical skeletons.

Abstract

We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a coarse ''subset'' of the Luna-Vust data of a spherical variety. Our smoothness criterion improves upon classical criteria by removing the dependency on external reference tables.