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

TL;DR

This paper investigates the structure of the moduli scheme of affine spherical varieties with a free weight monoid, revealing that its irreducible components are affine spaces and describing the tangent space at a key point.

Contribution

It provides a detailed description of the tangent space and geometric structure of the moduli scheme for affine spherical varieties with free weight monoids, extending understanding of their deformation theory.

Findings

The tangent space at the most degenerate point is explicitly described.

Irreducible components of the moduli scheme are affine spaces.

The scheme's structure is clarified under the assumption of a free weight monoid.

Abstract

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.