The moduli scheme of affine spherical varieties with a free weight monoid

We study Alexeev and Brion’s moduli scheme $M_{\mathrm{\Gamma }}$ of affine spherical varieties with weight monoid $\mathrm{\Gamma }$ under the assumption that $\mathrm{\Gamma }$ is free. We describe the tangent space to $M_{\mathrm{\Gamma }}$ at its `most degenerate point’ in terms of the combinatorial invariants of spherical varieties and deduce that the irreducible components of $M_{\mathrm{\Gamma }}$, equipped with their reduced induced scheme structure, are affine spaces.