Spherical varieties with the $A_k$-property

Abstract: An algebraic variety is said to have the $A_k$-property if any $k$ points are contained in some common affine open neighbourhood. A theorem of W{\l}odarczyk states that a normal variety has the $A_2$-property if and only if it admits a closed embedding into a toric variety. Spherical varieties can be regarded as a generalization of toric varieties, but they do not have the $A_2$-property in general. We provide a combinatorial criterion for the $A_k$-property of spherical varieties by combining the theory of bunched rings with the Luna-Vust theory of spherical embeddings.