The Moduli Scheme of Affine Spherical Varieties with a Free Weight Monoid: Table 1.

Abstract

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.