1 Introduction  
2 Notations  
2.1 Spherical embeddings  
2.2 The lattices \( M \) , \( N \) , and the colors of \( G/H \)  
2.3 The valuation cone and the maps \( \sigma \)  
2.4 Colored fans  
2.5 The classification of \( G/H \)-embeddings  
2.6 \( G \)-equivariant morphisms between \( G/H \)-embeddings  
2.7 Divisors, Cartier divisors  
3 Smooth, locally factorial and \( \mathbb{Q} \)-factorial varieties  
4 Gorenstein and \( \mathbb{Q} \)-Gorenstein varieties  
5 (log) terminal and canonical singularities  
6 Conclusion and example