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

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