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

The Moduli Scheme of Affine Spherical Varieties with a Free Weight Monoid