New and old results on spherical varieties via moduli theory

TL;DR

This paper explores the structure of spherical varieties using moduli theory, providing new proofs and descriptions of their geometric and algebraic properties, especially for saturated monoids.

Contribution

It offers a complete description of the tangent space at a fixed point in the moduli scheme for saturated monoids, and proves the freeness of root monoids of affine spherical varieties.

Findings

The tangent space structure at the fixed point is fully described for saturated monoids.

The root monoid of any affine spherical G-variety is proven to be free.

All irreducible components of the moduli scheme are affine spaces for saturated monoids.

Abstract

Given a connected reductive algebraic group $G$ and a finitely generated monoid $\Gamma$ of dominant weights of $G$, in 2005 Alexeev and Brion constructed a moduli scheme $\mathrm M_\Gamma$ for multiplicity-free affine $G$-varieties with weight monoid $\Gamma$. This scheme is equipped with an action of an `adjoint torus' $T_{\mathrm{ad}}$ and has a distinguished $T_{\mathrm{ad}}$-fixed point $X_0$. In this paper, we obtain a complete description of the $T_{\mathrm{ad}}$-module structure in the tangent space of $\mathrm M_\Gamma$ at $X_0$ for the case where $\Gamma$ 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 $\Gamma$ all the irreducible components of $\mathrm M_\Gamma$, equipped with their reduced subscheme structure, are affine spaces.