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Let $X$ be a normal projective variety with finitely generated Cox ring. Consider its characteristic space $p:\widehat{X}\rightarrow X$. This means that there is a torus $T$ acting on $\overline{X}=Spec(Cox(X))$, $\widehat{X}\subseteq \overline{X}$ is the locus of semi-stable points with respect to a suitable polarization $\lambda_X$ and $X$ can be obtained as the GIT quotient $\widehat{X}//_{\lambda_X}T$.

Now, for any polarization $\lambda$ we may consider the corresponding set of non semi-stable points $X(\lambda)^{nss}\subset\overline{X}$.

Assume that $X$ is a spherical variety. Does there exist a relation between the number of irreducible components of the sets $X(\lambda)^{nss}$ and the Picard number of $X$?

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