Spherical varieties with the $A_k$-property

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.

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Received 4 April 2013

Published 9 November 2017