New and old results on spherical varieties via moduli theory

Given a connected reductive algebraic group G and a finitely generated monoid Γ of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme MΓ for multiplicity-free affine G-varieties with weight monoid Γ. This scheme is equipped with an action of an 'adjoint torus' Tad and has a distinguished Tad-fixed point X0. In this paper, we obtain a complete description of the Tad-module structure in the tangent space of MΓ at X0 for the case where Γ is saturated. Using this description, we prove that the root monoid of any affine spherical G-variety is free. As another application, we obtain new proofs of uniqueness results for affine spherical varieties and spherical homogeneous spaces first proved by Losev in 2009. Furthermore, we obtain a new proof of Alexeev and Brion's finiteness result for multiplicity-free affine G-varieties with a prescribed weight monoid. At last, we prove that for saturated Γ all the irreducible components of MΓ, equipped with their reduced subscheme structure, are affine spaces.