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# Mathematical Research Letters

## Volume 24 (2017)

### Number 4

### Spherical varieties with the $A_k$-property

Pages: 1043 – 1065

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n4.a6

#### Author

#### 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 Włodarczyk 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.

Paper received on 4 April 2013.