On the irreducible components of moduli schemes for affine spherical varieties

We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $\Gamma$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_\Gamma$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_\Gamma$ to be irreducible and exhibit several examples where $\mathrm M_\Gamma$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_\Gamma$.