# GKZ decomposition for spherical varieties

If $X$ is a complete toric variety the GKZ decomposition of the effective cone $Eff(X)$ of $X$ corresponds to its Mori Chamber Decomposition, and therefore it encodes the birational geometry of $X$.

If, more generally, $X$ is complete spherical variety does there exist an analogue of the GKZ decomposition?

• Certainly the effective cone is still a rational polyhedral cone. In what terms are you asking for a description of this cone, e.g., when $X$ is the flag variety of Borel subgroups of a semisimple algebraic group? In this case the effective cone equals the nef cone, and this is a simplicial cone. But I do not see any obvious connection to the GKZ decomposition. – Jason Starr Jun 9 '16 at 13:38
• Just to clarify, are you asking whether spherical varieties are Mori Dream Spaces? Or are you asking for some combinatorial description of the chamber decomposition of the effective cone / moving cone? – Jason Starr Jun 9 '16 at 13:48
• Thanks for the suggestions. I know that spherical varieties are Mori Dream Spaces. I was rather asking for a combinatorial description of the chamber decomposition of the effective cone. – J.Phoenix Jun 11 '16 at 18:29